Well, first of all I should mention that I have the game on PS3 (which is the worst version, even worse than the Xbox 360) and I also have it on PC (which is of course the best version, mainly in terms of graphics).
First I should say that the game used to be a total mess.....bad graphics, the resolution was at 480p, it ran at 25 locked, and let me add that it has no campaign and it had no DLC either, and for zombies only 1 map. Then the game got an upgrade, they improved the graphics in the menu exactly like on PS4, but the images were running at 15fps, they also fixed the graphics as they raised the resolution to 720 and increased the fps to 30 but not locked (meaning there are quite a few framedrops + when the frames drop the resolution drops to 480 and then rises again to 720, something that tires the eyes).
They also added 2 DLCs for multiplayer and 2 for zombies (big plus).
To summarize:
1) The graphics, although improved, do not feel like a 2015 game + the framedrops
2) The game has a relatively small population, about 350 people per day, but if you own the DLCs it will be hard to find other players. Almost nobody plays zombies.
3) The lag is unbelievable....even on European servers you may be playing at 80ms and suddenly it can go up to 400, then 120 and back to 400 (you can understand how much you will struggle, especially in Greece).
4) One positive is that even if you are a noob, don't worry, your kills will be high since all the pro players have moved to PS4.
5) It does not have split screen so you can play with friends, which is a big minus
6) It does not have a campaign.
Compared to PC: the PC crushes it in everything, even in price, since you can find it for €9, which is what I bought it for. If you are a fan of the game, though, you have nothing to lose, you can buy it anyway because for a console price it is worth the money. I hope I helped and guided you to make the right decision

Since I am a fan of the Call of Duty series, I decided to buy Call of Duty Black Ops 3 for Christmas to see what it is like, because I see it getting a lot of criticism. I don't mind that it doesn't have a campaign because I am a fan of online games, although the campaign is a very important part of the game. Its graphics do not remind me of a 2015 game, but as soon as you see it, it feels like you are playing Call of Duty World at War, which came out in 2008, and it lags often. In general, it has fewer options compared to next-generation consoles like the PS4, so don't expect it to be a strong Call of Duty. But let's not forget that the game was intended only for PS4 and not for PS3, so if you want to have a complete Call of Duty Black Ops 3 with good graphics, you need to move to PS4, and Call of Duty Black Ops 3 is also the last Call of Duty released on PS3, meaning no more Call of Duty on PS3. But beyond that, its servers are not empty; that is, there are still people playing it on PS3. I did not regret buying it at all, and I play it every day. However, anyone who wants a good Call of Duty to play online with nice graphics and almost full servers should get Advanced Warfare, because it is my favorite. Rather than giving 50 or 40€ to buy Black Ops 3, it is better to buy Advanced Warfare, which costs about 25€, and you will also have money left over to get another Call of Duty like the legendary Modern Warfare 2 or 3 or Black Ops 1 or 2. Finally, there is no reason to criticize the game because, as you have understood, Activision is focusing more on PS4 and has left PS3 behind.