For Bryan to win in 1904, the Democrats would have had to nominate him for the third consecutive time: a rather blatant admission the cupboard was bare of all other talent, which doesn't bode well going in.
There is no way under the sun that Bryan could have beaten TR in a conventional two-party race: TR would have won by a monumental landslide, taking everything except the former Confederate states (and it's not out of the question that Texas, Virginia, or Tennessee might have defected); indeed, Bryan probably wouldn't have even carried his nominal home state of Nebraska.
If, however, TR were not in the equation (I recall a near-miss carriage accident that has been mentioned around here once or twice), then the Republicans would probably have nominated Charles Fairbanks or possibly Elihu Root (Albert Beveridge would have been a reasonable successor to TR but didn't have quite the prestige). Now the GOP has someone at the top of the ticket who's more in the prototypical Republican form, but it'll still be difficult for Bryan to win, given the relative prosperity of the times.
Still, for the sake of discussion, let's say Bryan is able to deliver the mother of all campaign speeches that sways a lot of undecided voters and allows him to win a squeaker of an election. What follows is not pretty:
* America's emergence onto the world stage is brought to a grinding halt. There's no Great White Fleet, at a minimum. Likely also appropriations for the Panama Canal are curtailed or eliminated.
* Prohibition is likely enacted about sixteen years earlier than in OTL, but with all of the attendant corruption and problems nonetheless.
* The Russo-Japanese war rages to a conclusion unchecked, with the Japanese in control of Port Arthur and much of what we know in OTL as the Pacific coast of Russia. This, in turn, could easily lead to destabilization of the Romanov regime about a decade earlier, with a revolution to follow.
* Race relations in the US, which had made some progress in OTL with TR, would stagnate and perhaps even go in retrograde motion, especially if Bryan placed one or more southerners in positions of responsibility.
On the whole, Bryan would probably rank with Pierce and Buchanan as one of the most ineffectual and worst presidents in US history. And like Pierce and Buchanan, it would take a powerful GOP administration thereafter (say, Charles Evans Hughes beginning in March 1909) to undo all the damage and get things moving again.