I haven't played 5 so I can only go by what I've read on discussion forums.
From what I gather, the story in 5 is a little more "what if" style alternate history than found in the previous games but not outrageously so.
For game play, the big changes are:
Invasion mode. Other players can drop into your session and take on the role of an enemy sniper trying to hunt you down while you play through a mission. Reception on this seems mixed with a fair number of complaints about people behaving as you'd expect, as in they are being asshats and just doing what they can to ruin your day by trying to break the game.
Character progression and customization. You can add certain skills and perks as you gain XP. Stuff like faster reloads, tossing back grenades, better stealth, etc.
Weapon customization. They removed a lot of the old standbys and went down the Battlefield route by adding in a lot guns that were either very rare or only on the drawing board. They also added in a lot more customization options for most firearms, much of it ahistorical (Stens with IR sites, 1911s with scopes, buttstocks, and gangster grips, silenced MP40s with subsonic ammo, etc).
Non-lethal melee takedowns. No idea why they added this in.