The hard drive matters. Lots of games want to install stuff onto the HD, so in reality many 360 games don't work on the 4GB version because it simply doesn't have the space to install a single game. So if you buy the 4GB version there will be a ton of games that simply won't play on that version. Though this might be overcome by using a USB memory stick... I think they allow up to 16 or 32 gb sticks. But you'd have to buy one and still have very limited space and have to manage that with every game or two you install.
I also wouldn't buy a 360 used. Since they are such low quality design, odds are high you will get one that fails in short order. Especially if it's one of the earlier designs, and they had like 4 of them or so.
The Xbox 360 is made like garbage, even the newer 360's have issues, but not as much as the earlier versions. But when my 2nd disc drive was going out on my launch console (after RRoD, disc scratch, hard drive fail, and before another RRoD), I couldn't install Halo 4 onto my hard drive, and thus was never able to play Halo 4. So if you want a 360, definitely get one with a good sized hard drive. At least 120 GB's. But ones that size are probably older, so you should probably go with the 250 GB's.
Another thing to take into account is that for this time in the life cycle of the console, it should be MUCH less expensive. So you would be paying a huge premium. Think of it this way, the Xbox One literally has a sub $100 PC graphics card inside of it ($60-80 worth), and a laptop CPU, and they are charging $399-499 for it. So the fact they charge $249-299 for a 360 is highway robbery. A $29.99 PC graphics card is better. In all honesty, the guts (power) in the Xbox One show it shouldn't have ever been priced above $299.99. It's a stretch for their new console to be charged above that. You could buy a graphics card twice as powerful as the Xbox One, a year and a half earlier then when the Xbox One came out for $100 less. The PS4 is also underpowered, but it's graphics components are a class above. (Which is one reason why a $399 more powerful system is easily outselling a less powerful $499 system where the makers gave the middle finger to gamers with all their idiotic ideas to take stuff away from you). So basically you should realize that a 360/PS3 price cut could easily happen this year. You might find come Xmas these older consoles at $50-100 cheaper then they are now.
Also be aware that any online gaming on 360 will require an XBox Live account and subscription at $60 a year. They also like to bill it recurring. Buy the cards in store to get around that issue, and sometimes you can get a discount too.
Xbox One and PS4 both have a subscription needed to play online, however Sony doesn't keep as much behind the paywall. For instance, free to play games, on the PS4, are actually free to play, even if online. Netflix and others apps aren't behind the subscription paywall. Meanwhile Microsoft is more restrictive, though they are trying to change things a bit, but not sure how exactly. Mostly because PS+ is much better value, because it gives you 2 free games a month, which Microsoft has now copied with Games with Gold.
Meanwhile the PS3 has NO subscription to play online. Oh, and if you get a PS+ subscription you get free games on PS3, PS4, and Vita handheld. Great deal.
The Xbox One was designed to not work unless it could check in online every 24 hours. Luckily people who don't like to bend over, complained big time, and got idiotic stuff like this changed, but of course, only after a huge sales disparity between the Xbox One and PS4 showed up. Their Xbox One UI is horrible, because they assumed everyone would want to use their voice, they made it really crappy for people to navigate with a controller. Basically forcing you to use voice if you didn't want to go crazy. This will be updated, I'm sure, but when and what year is unknown.
FWIW many games want to EACH install 6-10 GB's on an Xbox 360 hard drive, and for PC games, many games are 20-40 GB's and are on their way up to much, much higher. Like 100+ in a couple of years. But that's PC using higher quality assets that the 360 won't have. But PS4/XB1 will get assets decently up that chain as well.
Even 500 GB's on the new consoles is really, really small. Again if digital download games are ever going to be bought, even Xbox One and PS4 will have issues holding many games, aside from the small indie ones. Digital downloads you can't sell or trade the game, but you can get it much faster and don't have to worry about scratched discs or your kids friends stealing the discs. You can get the games on day 1, rather then wait until it's in stock or the UPS man comes a week after launch, unless you pay the insane overnight shipping charge. So while you may not want digital downloads now, you may want to in the future, and these above issues are most of the pros/cons behind it to ponder for the future.
Depending on size of the game both new systems will hold 10-15 full size AAA games at 15-40 GB's per game. There is talk of them increasing the size to 1 TB, which is still REALLY REALLY SMALL. If people want to go the whole generation 4-8 years, and buy mostly digital games, they are going to need 2-6 TB's of space. Hell my steam account has probably 4 TB's taken up, and almost half my games aren't installed, and again, the size of games is set to ramp up drastically. The new Wolfenstein was like 40 GB's.
So to fit the same amount of games on a 360, a 250 GB 360 is equivalent roughly to if they offered a 1.5-2 TB version of the Xbox One. So a 360 with a 250 GB hard drive will hold quite a few games. Still smallish, but definitely workable.
Luckily Xbox One allows for a external HD to be hooked up via USB. PS4 allows you to remove the internal hard drive and put in a new internal HD...though it has to be a notebook one or a really thin desktop one. Somewhere between 11-14 MM thick...can't remember what exactly. So you can replace the PS4 one, but can't use an external one. The Xbox One can have an external one, but can't replace the internal one. Oh and the PS4 can use solid state hard drives, which vastly speed boot and loading times...but are REALLY expensive.
PS4 does have a camera you can buy separately. Both next-gen consoles barely use the camera for gaming. Sony advertised it far less, and hasn't specifically made games for it. Microsoft hyped it and forgot to bring the software. They did make Kinect Sports Rivals, and I think a couple of other things came out, but motion gaming in general has been determined to be a fad, and most gamers HATE motion gaming. So I don't expect much to be there in the future.
So if you are looking for the biggest library, it's in the past and would be located with the Wii/360/PS3. But the Dance Dance Revolution games should be fine and continue on. They seem to be doing well.
The Kinect is mostly about voice commands these days, and really they could of just used a $2 microphone to achieve this.
Now that they've shifted their focus, removed Kinect from the box, and the main internal Microsoft studio that made Kinect games for them is going to make non-Kinect games, Kinect as a whole won't probably be supported much, gaming wise. Of course 3rd parties will like some voice controls and leaning in say Battlefield games, but they will generally be similar across platforms, and thus PS3/PS4/Xbox One/360 will get similar experiences. Like Battlefield.
The Sony Move on PS3 worked REALLY WELL. By far the most precise in the movements. It could literally pick up the spin of your arm to put spin on the ping pong ball, and worked EVERY TIME. Unfortunately it didn't sell well. But some of this is used for the PS4's camera, which is why the PS4 controller glows like the ball on the Move controller did for the PS3.
There is also the debate about how much information such cameras are collecting. Microsoft IS in bed with the NSA, and have been since the 90's at the very least. One of the Snowden documents actually showed they were investigating how to get access to the original Kinect on Xbox 360 a full YEAR before it even came out. I know most people think there's nothing to see, but that's just bull. There are plenty of reasons they've come up to spy on ordinary law abiding citizens, and in fact that what they are looking for, and the whole program is geared for, instead of terrorists. That and blackmail, corporate espionage, etc, etc.
So on that note, the Xbox 360 camera seems to be 'safer' then the Xbox One camera, because it's far less powerful, doesn't have it's own CPU, unlike the Xbox One version, so on and so forth. But the fact is the companies are required by law NOT to tell you what they are doing, so in fact they WILL LIE to you, because they HAVE TO, by law.
You shouldn't trust Sony or Microsoft, but at least Sony wasn't plastered on the Snowden documents unlike Microsoft which is basically numero uno in bed with the NSA.
The PS3 and 360 won't be supported much longer, though Sony usually supports their older systems longer. In fact last year Sony continued with huge AAA releases even though they were coming out with a new system, meanwhile Microsoft shafted the 360 owners by switching their production to Xbox One.
Sony also has multiple times the number of studios compared to Microsoft, so besides getting the best console version of all multiplatform games...i.e. Activision, Electronic Arts (EA/EA Sports), so on and so forth, they will also get the most in house exclusive games...by far. So the best running Call of Duty or Battlefield will be on PS4...console wise. It should also have the biggest online base due to higher PS4 sales. Currently worldwide it's about 2:1 ratio of PS4 outselling XB1... trending larger.
Both the PS4 and XB1 seem to be better designed reliability wise the 360 and PS3. Though PS3 was decently made. The 360 was a nightmare of epic proportions. Neither consoles have a major issue like last generation. For instance, of the launch consoles, it is estimated that over half of all 360's from the first year failed within the first two years. Microsoft had to do a billion dollar charge to pay for the repairs, and that was just ONE of MANY issues that POS console had.
Neither the Xbox One or PS4 are backwards compatible. Meaning you can't play PS3 or 360 games on them.
The Wii U doesn't have as much motion gaming... BUT it is backwards compatible. I never had a Wii, but I did buy a Wii U, so I bought some Wii games to play on it.
It is cheaper, but it is also much less powerful. The main cost is the gamepad which has a 6 inch screen on it. It's cool, but they don't use it very well. But it is pretty cool. I loved calling killstreaks down with it in Call of Duty. So power to price it is even a worse bargain then an Xbox One. BUT Nintendo has the Nintendo IP's, like Mario, Star Fox, Zelda, etc, etc, etc...they put out a ton of good games no one else can. But 3rd party support has dried up big time.
Both the PS4 and Xbox One have blu ray players, so you get one if you don't have one by now anyways. Of the older consoles only the PS3 has a blu ray.
As for 360 vs PS3 power wise... 360 had better running multiplatform games due to easier coding, but the PS3 had more power but almost impossible to code. So first party exclusives look better then the games on the 360, but multiplatform games always ran better on the 360.
Also don't forget about PC. These days a PC connects to your HDTV, so you can literally have a PC hooked up to a PC and PS4 and Xbox One and Wii U all at the same time. Also most PC games that have console versions can use a controller. Hell I use a 360 controller for windows on many of my games. You can tailor what you need and upgrade as necessary and never pay a premium price if you put it together yourself (also pretty easy these days). It'll cost similar or more, depending on parts, but it's still a great value because it's a PC, and with all the sales on Steam, GoG, Greenmangaming, Origin, Uplay, etc, etc, etc... you get games MUCH CHEAPER on PC then on consoles.
Every game you purchase on a console sends $10 to Sony/Microsoft/Nintendo. Now some devs scumbag you and charge $60 for a PC version, but many only charge $50, and it'll be the best version to boot. Steam sales are epic. Tons of games for $3-5-10-20 that are normally $20-60. So you can get a lot more games for your money on PC then anywhere else.
Sorry for the length, just since you said you've been out of gaming for a while, it's important to see the entire spectrum before you decide to buy something. Because as I've said, you could get the power of an Xbox One on PC in 2010 on a single card for less then the Xbox One cost. Let alone the ones who want to put four $1500 graphics cards in their system. You can do that, or buy a graphics card that has 2-3 more raw power for the price of an Xbox One with Kinect ($400-500 or so for the graphics card). Literally there are PC's out there that are 25x as powerful as an Xbox One. 15-20x a PS4. But the average midrange PC that will cost 800-1200 depending on how selective you are can easily double or triple the raw power of an Xbox One... and of course every year or so a new line of graphics cards comes out, like Nvidia's Maxwell later this year (but the true 20nm newer smaller lithography process ones worth buying won't come out until next year, the initial ones will be on 28nm).
So overall you have tons of things to ponder. Or maybe you are going to want two consoles. A newer one and an older one. Because in the end, you want what you want and what will make your kids happy. Hopefully the above info gives you what you need to make the best decision. Just realize it's quite complicated based on everything that revolves around gaming today.