As someone else said, it would be hard for Roosevelt to win without winning the GOP nomination. The only way that Roosevelt could win on the Bull Moose ticket would be if the Democrats nominate a more conservative or regional candidate like Champ Clark or Oscar Underwood. Even in that case the Democrat would still be favored in the Electoral College since they would hold the Solid South and probably take much of the Midwest and Mountain West with pluralities due to the division of Republican votes.
If Roosevelt somehow wins on the Bull Moose line, I suspect the Bull Moose Party would still have ultimately faded. The Progressives were really just a vehicle for TR. Either TR reconciles with the Republicans and you get a Progressive-Republican fusion by the 1920s or the party fades once he leaves the scene. Maybe it has some vestigial support in certain areas like New York, Wisconsin and California and limps on for a few decades. Alternately - though unlikely - maybe the Progressives fuse with the Democrats, which might lead to an earlier shift of the Democrats towards progressivism.
In terms of policy, a TR presidency probably is similar to the Wilson Presidency. We might, however, get national health insurance. In foreign policy, TR probably intervenes more forcefully in Mexico on the side of the right and probably enters WWI earlier.
The long-term partisan consequences are harder to predict. If TR is elected as a Republican or if the Bull Moose Party reconciles by 1920 with the GOP, then Democrats are the beneficiary of postwar "return to normalcy" sentiment. Which means they might be in power when the Great Depression hits and Republicans may return to power as activists, becoming the party of modern American liberalism.
If, on the other hand, Roosevelt, elected on the Bull Moose line, remains estranged from the Republicans, they probably return to power after his second presidency and the long-term outlook is actually fairly similar to real-life.