Well, one thing's for sure. Rome will be almost impervious to barbarian incursions. They'll lack the infrastructure and resources to build steam engines. Lets assume that the Romans go for applying this steam technology to transportation, with steamships and eventually locomotives.
With steamships patrolling the rivers of the Empire, the Romans would be able to move the Legions much quicker than by foot. With locomotives, they'll be even faster. The key factor is speed. Increases speed allows for increased mobility. You can outmanuver your opponents. Then, you've got the added advantages of better logistics.
There are steam weapons that could have been developed, weren't there? So, now we've got increased firepower and increased mobility.
Now, with all these transportation advances, commerce will get a boost. Foodstuffs can be shipped faster, more reliably. Plus, this will see an increase in the demand for fuel. Wood, possibly coal. Either will expand their respective industries.
Then, if coal becomes the dominant fuel (as it likely would), there will be an increase in mining activity. One of the first uses of steam engines was in mines, as pumps. They'll likely be applied eventually there.
From this point, one would assume that people would realize all the other uses to which steam power could be applied, though perhaps not.