I've watch both versions several times. John Wayne plays Cogburn with a glint in his eye and high spirit, Jeff Bridges take is grim and foreboding. Each actor really owned the part in his own way. Glen Campbell vs. Matt Damon is no contest, Damon is an actor by trade and was much better.
I though that both Robert Duvall and Barry Pepper did equally good jobs playing Lucky Ned Pepper.
The ending of the Duke's version in happy and the ending of the Bridge's version, which follows the Portis novel is more realistic, though much more somber.
Conclusion- the John Wayne movie is more fun to watch and therefore my favorite.
P.S. John Way deserved an Oscar for The Searcher's, She Wore A Yellow Ribbon and The Shootist. True Grit, well Hollywood owed him one for hosing him for Ethan in the Searchers.