Hi LiJ, I'm still tryin' -
If you're speaking of a personal committment / dedication, yes, it's fine to treat them as equivalent.
As GG pointed out, the other meaning would be if your company assigned you to work only / solely / full-time on that project, your time would be "dedicated" to that project in a sense that "committed" would not accurately describe. Still, on the other hand, if it were a "committment" and your friend asked you to help him out with a different project, you might reply, "Sorry, my friend, I'm committed to finishing this project of mine by the end of the week, and I just won't have time for anything else." This could refer to a promise to yourself or a promise to your employer.
Kind regards, - A.