In this context, "falter" means "to weaken" or "to waver", but "fail" is a more final term, meaning "to be unsuccessful". For example, a team might falter, but succeed in the end. If they fail, they lose in the end.
Falter is one of my favorite words. It is often applied to circumstances where someone who is a favored player in a sports' contest fails to perform the way he/she should. Its equivalent, using the word "fail," would be, eg., ". . . so the hero failed to come up to his abilities and the team lost."