Edit: Stu. Im on mobile.
It depends who you ask. Both exits were influenced by a variety of different factors, and both no doubt felt pressured to leave at one point or another. I think it would be fair to say that Stu Sutcliffe eventually decided to do so voluntarily and move on to other things he enjoyed more, where as Pete Best's treatment would be called more like a firing at best.
Both faced issues of talent. Stu Sutcliffe was very rudimentary in his style, even by the standards of the Beatles (who were all fairly rudimentary on their various instruments at that point in their career, circa ~1960). In particular, he rubbed Paul McCartney the wrong way. Paul pushed Stu to improve his playing, and privately pushed other members of the band to toss him as well, but it never really came to that because Stu was the only one willing to play the bass. It was understood among the them that George was the best guitarist and that John, as de facto bandleader and source of so much of their material at that point, was in no danger of losing his role as rhythm guitarist. McCartney, also playing rhythm guitar at that point, had no desire whatsoever to move to bass, as he knew he'd have to do if Sutcliffe left. Despite that, he was no fan of Sutcliffe's. He resented how close Sutcliffe was to John (they were old school friends) and how influential Sutcliffe became during their residencies in Germany (it was Sutcliffe who first got the "Beatles" haircut, adopted the German style suits, etc).
Sutcliffe was aware of the tension in the group and under no great delusions about his own skill when it came to music. Several voices in his life were telling him that he should give up music and pursue art, which he had studied in school and which had long been his first passion. In '61 he agreed that it was for the best and left the Beatles, passing his Hofner bass (another Beatles icon) off to Paul and stayed in Germany, focusing on his art until his death the next year. As an aside, John has been accused of causing the injury that ultimately led to his death, but it's entirely inconclusive.
Pete Best was hired before the first Hamburg residency because the Beatles didn't have a permanent drummer, so they auditioned Pete and were sufficiently satisfied to bring Pete along. From the outset he was distinct from the group. He never matched the visual style of others, either their rockabilly look (denim and leather and skiffle haircuts) or their more familiar Beatles look (the mop-top hair and matching suits) which caused friction between the members and would become an issue for Brian Epstein as well. He participate in the drug culture the other four adopted in Hamburg either.
Similarly to Sutcliffe, he was the object of some jealousy. It's widely held that when he was in the band he was the most popular member - he was a good looking guy and was definitely distinct from the others. However, the biggest issue was ultimately his ability. When the Beatles were getting popular enough to start recording singles, Best played a session with them and was by all accounts a severely limiting factor. His beats were shallow in their simplicity, and despite their basic "four on the floor" style he had difficulty actually keeping the beat, which often caused the whole group to collapse. He had also become unreliable for live performances, which irked Harrison to no end (he took responsibility for encouraging his dismissal). Ringo was the hot drummer in Liverpool at the time, seen as the best drummer in town and playing with another popular group, Rory Storm and the Hurricanes. They wooed Ringo into coming and playing with them and soon adopted him as their permanent drummer, a move that was very unpopular with a lot of their fans at the time. Brian Epstein was tasked with letting Pete Best know that he was out, somewhat without warning.
So ultimately they were fairly similar cases. Sutcliffe departed voluntarily amidst some tensions surrounding his ability and relationship with the other members to focus on where his gifting more naturally lay. Best was ousted amidst greater tensions surrounding his ability and different aspects of relationship with the other members.
Sources:
Consider browsing the wiki articles on these guys, which are all pretty good. Norman's book is also thorough, based largely on various interviews with participants in the events (though few with the Beatles themselves), and enormously readable.