They're similar, but they don't quite mean the same thing. To gather oneself = to prepare oneself mentally to do something that requires effort For example: He gathered himself before entering and delivering his speech. To pull oneself together = to become calm and regain control of one's emotions For example: Jack was in shock at Jane's behavior but he finally pulled himself together and managed to give her a smile. While it is possible to say "gather himself" with or without "together", "pull himself" on its own is incorrect. So all these sentences are correct: Fortunately the short delay gave him time to gather himself. Fortunately the short delay gave him time to gather himself together. Fortunately the short delay gave him time to pull himself together.

"To pull himself" is wrong, and "to gather himself together" is very awkward/mostly wrong. I personally wouldn't use the second one, although it's more acceptable than "to pull himself". The correct phrases are "to pull himself together" and "to gather himself". And both of those correct phrases are, in fact, mostly interchangeable. There's no particularly pertinent difference between the two; the only thing I might say separates them is that "to pull oneself together" can denote that their current condition is more extreme. For instance, if a person lost some papers during a business meeting and got momentarily flustered, you could tell them to gather themselves. If they're in the same situation, but they've broken into hysterics, tears rolling down their cheeks, and they're loudly cursing about how terrible their life is, then you'd say "pull yourself together!". Using either phrase in either case is acceptable, but it's slightly better to use "pull oneself together" when a person's in a really terrible condition.