I like most of the Marvel characters but I haven't kept up with comics in a long time outside of occasionally glancing through a Wizard magazine at the store, so I had no idea what happens in either storyline. Here is OXM's synopsis of each:
Secret War (2004-2005)
This Marvel miniseries (not to be confused with the 1980s crossover Secret Wars) had several street-level superheros -- including Spider-Man, Wolverine, and Luke Cage -- dealing with the messy aftermath of their invasion of Dr. Doom's country, Latveria. Why was it "secret"? Former SHIELD head Nick Fury had brainwashed the crimefighters to forget the illegal incursion.
Civil War (2006-2007)
Continuing the post-9/11 theme of security-vs.-freedom, Civil War involved mandatory registration of metahumans after a crowded school was annihilated in a super-battle. The Marvel heroes split into two factions -- a pro-registration group led by Iron Man and an anti-registration side led by Captain America; the miniseries ended with Cap being arrested (and, in his own comic, slain by a villain-controlled ally).
Due to the game being based loosely on these storylines, it means the roster of 24-or-so characters will be somewhat trimmed for most of the game, but developer Vicarious Visions said "most" of the characters will be available to you no matter which way you go, and there will be at least 2 possible endings.