What happens to armor a marine has gained during a campaign when he dies?
I'm not a player of the Doom board game, but I'm a player of many other board games and the Doom computer game and therefore think I can make a fair interpretation as to what the rulebook and campaign book rules might mean together.

I think rule 1 (A killed marine...) means that at any time the marine is killed during a scenario he will respawn with the default (as in without any items and bonuses) armour and health.

The second rule (A marine starts a scenario...) in my opinion is not contrary to that because it refers to the beginning of a scenario. 

If a marine is killed during the scenario then rule 1 applies and rule 2 does not, if it is the beginning of the scenario rule 2 applies. This seems to match with the Doom computer game, if you finish a level you can take the armour and life to the next level, however if you are killed during a level you respawn with the default armour (0 I believe) and the default health (100 I believe) at the beginning of the level.