In the comic we have mostly seen Captain America as the leader of the Avengers. He is the tactician and despite being less powerful than other members has been one of the teams greatest assets. Has this carried across to the MCU? ...well not really.
The Captain was mostly sidelined in the Avengers to make way for Robert Downey Jr. Now I love him as Tony Stark and he has made a huge contribution to establishing the MCU but just because he has become popular that doesn't mean he should be elevated above the other characters. He wins all the arguments and makes the Captain seem like an idiot as does most of the movie. It is also Iron Man who saves the day whilst Captain America just punches a few Chitauri which makes up a tiny fraction of the attacking army.
We don't want Steve Rogers to become like Cyclops in Fox's X-men, pushed aside by a more popular Wolverine. He should be leading the team, coming up with the plans, holding his own bantering with Tony and kicking some serious ass. The character is awesome in The First Avenger and looks like he will be again in Winter Soldier. Yet in the Avengers he is just nearly as dull as the aforementioned Cyclops of the X-men movies (not quite, I mean that character adaptation was terrible.)
In my personal opinion Captain America should be more serious than the other characters which Marvel studios have appropriately done. It is just a shame that his introduction to the modern world was glossed over. This scene should really have been included:
http://www.youtube.com/watch?v=oulqXofUTms
It is the things like this which the MCU is lacking. I love their movies especially The First Avenger and The Avengers but they do need some deeper scenes like this. We don't want to go see a one of their films to find it has become nothing more than Superheroes punching each-other and then saying one-liners.
Also as a side-note I would like to say that I much preferred his costume in The First Avenger to in The Avengers. Winter Soldier is an improvement but I really like the one from the first movie.