Dragon Ball Z: Battle of Z doesn't really offer anything new as far as how it tells the Dragon Ball story. There have been games that tackle this task in everything from strict adherence to the script to a board game style progression even to the point of telling alternate timelines of events.
Battle of Z finds a middle ground by sticking fairly close to the actual events, but putting people in different places where it wants in order to tell a more entertaining story or just to add more bodies to a particular fight. That's because what
Battle of Z adds to the license is the ability to fight 4v4 battles in huge arenas, and there are simply some fights from the storyline that would have felt like a step backwards in this game if it had to stick closely to the original text.
Battle of Z handles the series' story by breaking each saga up into a series of missions. Each mission progresses the story a little more, but some run parallel to others and some fit completely outside of the primary timelines (i.e. the movies or even scenes acted out as the manga portrayed it). While you don't need to clear everything in a saga in order to move on, each mission you do finish can unlock characters, doles out experience and can reward the player with cards.
These cards are Battle of Z’s form of customization, a long standing feature of DBZ fighting games. Each character has slots next to 6 different categories, and cards fill up these slots to give different stat boosts. The cards, of course, come with various restrictions like level requirements so just because you were rewarded with a powerful card early in the game, it doesn't mean you can put it in play right away. You might have to wait until you have a character that actually qualifies to use it.
As for the four-on-four combat, the idea behind having your allies on the field with you is that you can call on them for help, they can attack other opponents while you are busy, and you can even pull off multi-person combos as you throw an enemy from one side of the field to the other in a messed up game of hot potato. That being said, I found myself handling most fights as if I was the only one on my side and felt a nice surprise whenever one of my allies happened to be in the area to lend me a hand. On the flip side, I found myself slightly annoyed when an ally would call for help and I had to break away in order to give him energy so he could continue doing whatever he was doing on his side of the arena. Granted, this might not have been the best strategy, or not necessarily how the designers intended the fights be played, but it's how they always seemed to be for me.
Dragon Ball Z: Battle of Z offered a few more options than just it's single player Story Mode. For one, a player having trouble getting through a mission with bots for allies could bring in some friends and help progress the story that way. It's all online though - Battle of Z doesn't feature a split screen co-op option. Quite frankly, this is one of the few drawbacks that I found in Battle of Z. While having the entire screen to myself is great - it would have been nice to be able to have a friend sitting next to me to play instead of going online and looking for a friend that happened to have Battle of Z as well.
There are also several online non-story based modes like 4v4 Tournaments or even an 8 player Free for All Mode. Another game has the two teams scouring the field for dragon balls as points and another is essentially classic team Deathmatch - just as a fighting game rather than an FPS.