My understanding is that many thousands of freemen joined the North's armies. How important were those extra soldiers to the outcome of the war?
Probably not very, for a couple of reasons
Black troops weren't sent into combat all that often, and not until late in the war. Off the top of my head, I can think of Fort Pillow, Fort Wagner, Battle of the Crater...and that's it. There might be a few more, but as a general rule, black troops were preferred for labor duties instead of combat. This basically continued all the way to WW2--some black units would see combat, but most were put into logistics or labor.
Armies in the U.S. civil war were very large (50-100k) and the addition of a few hundred here or there would probably not make much difference. And it would have made even less difference for the Federals because...
Throughout the war, U.S. armies were generally larger than their C.S. equivalents. This difference becomes especially pronounced towards the end of the war, when the South's lower population could not keep pace with the number of men the North could recruit. By 1864, U.S. armies were nearly double the size of the opposing C.S. forces. The addition of another few hundred/thousand troops on the Federal side wouldn't have changed much.
IIRC, the most significant groups of freedmen ended up being freed during Sherman's Atlanta Campaign and the March to the Sea? I don't know for sure--somebody with more knowledge can correct me--but I'm fairly sure Sherman did not just admit untrained freedmen into his Army and hand them a musket.
Sources:
Shelby Foote, The civil war: a narrative
Richard Slotkin, No Quarter: the battle of the crater, 1864.