I would say Rooney is the better all round player with Shearer being the better finisher. If I was forced to choose one than it would be Rooney. He has more to his game than Shearer did, he can play from wider role, off the main striker or as the main striker. He can bring others into play and he does score relatively a good number of goals. Shearer on the other hand was very much a old fashioned No.9 who did the job very very well. His goal scoring record is amazing especially as he played in sides which weren't very good though he won the title at Blackburn, he wasn't playing with great players. Sherwood and Wilcox were decent but not amazing whilst at Newcastle, half the time he player with bloody good players (Ginola and Ferdinand at first and than Bellamy and Robert under Robson) whilst the other half he played with some useless shit. And no he was still a consistent goalscorer. But he did also have good all round play, his partnership with Bellamy was evident of that where they really complimented each other and IMO were the best partnership in the league for a period.
However I think its important to remember certain factors, Rooney at Man United has consistently played with better players with understandably make him a better player whilst Shearer at times hasn't had that. In addition Shearer's game had evolved from being reliant on pace to playing his back to goal, an evolution which he achieved successfully, Rooney hasn't reached a point where he needs to do this and he's still relatively young. Also Shearer was able to come back from several serious injuries and was still be lethal whereas Rooney has been relatively fortunate in the injury department.