No, you don't have to play older ones. These are military shooters, so they assume you're new to the series except in the story, which is rarely ever good to begin with.
Anyways, I haven't played either, but here is a basic run-down of the two series' core mechanics:
Call of Duty is based more around smaller maps, smaller teams, and individual effort. It runs faster than most other games in the military shooter sub-genre and tends to get its excitement from this faster paced nature. Teamwork is a hit-or-miss. Well organized teams will always do better than disorganized ones, but teamwork is sort of pushed aside in the series, as doing good generally focuses on your own ability to navigate the map or control a section of the map, and you'll do that independent of your team 99% of the time. Class selection is entirely up to you and there are very few limits built into the system of what you can run. The game also has killstreaks, which you can think of as the worst idea known to man. Along with encouraging lack of objective play no matter how hard the developers try to fix this, it essentially rewards players who have already proven their better with a way to dominate even more. That's right, so the game isn't just unbalanced in favor of who wasted most of their life on it, but it is also unbalanced in favor of those who are clearly better. Sure, a game with good balance won't hinder those people, but it shouldn't excessively reward them, which is what CoD does far too often as it continues breaking everything that made CoD4 good.
Battlefield, on the other hand, is built around larger maps, larger teams, and group effort. It's slower than Call of Duty, and there will be times you may go a minute or two without seeing anyone only to get sniped and have to respawn without ever having seen any action. The game isn't infantry-focused like Call of Duty, but instead if focuses on allowing multiple forms of infantry and vehicular combat. The game features many vehicles that in Call of Duty would be killstreaks, and they obviously fit better and are more balanced into the mechanics than in Call of Duty, but if you don't like vehicles, then these can be a real pain to deal with. The series is also far more technologically advanced than CoD is and has a much more powerful graphics and physics engine. IW likes touting that their game runs at 60 FPS, but at this point, so does Battlefield, and if you're on the PC, you can take it even further with Battlefield, which I'm not sure you can do with Call of Duty. Team sizes are also 64 players as opposed to 16, and you'll split into squads of varying size, but this is really necessary on the larger maps. It does encourage more team-based play, but this can also make the game more annoying if your team doesn't play together, since individual effort isn't as easy to work with as in CoD. The games also run in dedicated servers, which makes the online theoretically more stable than CoD, especially in the long run.
Anyways, at this point, I would probably go with Battlefield 4. If you had asked this back in 2007, I would easily say Call of Duty, but the IW CoD games have fallen tremendously over the years. Not to mention, from the sounds of it, Ghosts just isn't a well designed game at all. BF4's biggest problem now is all the glitches and stability issues, but DICE has a very good history of supporting their games, so all the problems will be fixed in the near future.
