There were some real differences in the Presidential philosophies between President's Taft and and Roosevelt. First we have President Roosevelt's philosophy that a President could, and should do everything in order to serve the people of the United States as long as the Constitution does not specifically forbid it. President Taft on the other hand believed that the presidency is equal in importance to the rest of the branches of government and the President should only exercise powers that are directly stated and enumerated in the Constitution.
The pluses to Roosevelt's theory is that he could be a very involved and vocal president. He could be a President that gets shit done. Roosevelt could use powers that aren't specifically denoted in the Constitution and use them to unilaterally to advance his agenda. Unfortunately, the downsides include mostly a very very thin stretching of the Constitution and a possible public backlash for being much too strong and being "out of touch".
Taft on the other hand, had a few pluses as well, including the fact that he allows himself to work within the check and balance system. There would be a lot of advising between the government branches going on and the government would probably be making compromises left and right. Negatively however, Nothing that Taft truly wanted to get done would probably ever get done because the power specifically enumerated to the President is few and far between.
My personal thoughts on this matter are more in line with Roosevelt although on a few issues I agree more with Taft. When it comes to Legislation, I believe that the President should have a bit more of a role in attempting to pass bills because a President's agenda and how he gets elected is big face of our Government and I believe the President doesn't have enough power to fulfill his promises and enact his agenda. When it comes to military control however, I believe that the President should exercise less power in their control. Troop usage should be primarily left up to a wide body of people and voices (in this case, Congress) and the President should stick to his constitutional duties under this issue. My overall view then perhaps is a hybrid between the two view points as, on most issues, I would really prefer a very very strong executive but on a few issues, especially where lives of people are at stake, I prefer to have a weaker executive in these areas