Actually, Rutsey (original drummer) dropped them, right before Rush was to embark on an extended U.S. tour.
Basically Rutsey had done a few things to upset Geddy and Alex in the past, such as not submitting the lyrics for their debut album even though he was in charge of writing them. He kept saying he wasn't happy with them and when they were in to record the album (on a very limited budget) and Alex and Geddy finally demanded them as Geddy was supposed to start singing them that day, they learned that Rutsey had torn them up because he wasn't satisfied with them (Geddy has said Rutsey had "moments that were not strictly logical" lol, I wouldn't have been so charitable). So Geddy and Alex had to try to write lyrics asap (they wrote them in 2 days) so they could record the album in their allocated studio time.
Then there was the fact that Alex and Geddy wanted to take the band in a more progressive direction muscially while Rutsey wanted to play more traditional blues based rock and roll. He wouldn't compromise.
But Rutsey was the one who really ended it. When Rush got signed and more popular in Canada (with the release of their debut album in March of '74 and touring done behind it) he wasn't really happy as he wanted to keep the band more local/small. To be fair, he had diabetes and simply couldn't do the extended (later VERY long) worldwide tours Alex and Geddy wanted to in order to become really truly big.
So what happened, in the traditional telling, is that right at the start of Rush's first extended tour in the States (which was to start in Aug. of '74), Rutsey ended it, stating that he was bored/tired of playing with the band/the touring (which was 5 months long at that time). Alex and Geddy had to find a replacement drummer in less than 2 weeks (July 26, 1974 was the last Canadian date played with Rutsey; August 14, 1974 was their first date scheduled on a lengthy US tour to coincide with the debut album's release in July of '74 in the states) and happened upon Neil in the quickly held auditions. The rest is history.
But recently, in the Rush documentary (which you HAVE to watch, big fan or not...non-fan/casual fan friends of mine have watched and liked/loved it), the story is told a bit differently with different figures saying that John simply wasn't taking care of himself. He was touring with Rush in Canada (from March to the end of July in '74), which wasn't the best thing to do with his diabetes, but he was drinking heavily on top of that. Basically management and Alex and Geddy decided he had to go for his own health; they were sincerely worried that he would die if he continued what he was doing.
I think Rutsey left due to a combo of all these things: Rutsey's health, his disinterest in the sound Geddy and Alex wanted to pursue as well as becoming a really big band (or at least touring a lot: it seemed he wanted to just record/release albums and play local shows...not exactly the way to get worldwide success), and his refusal to be a "team player" at times.
Whatever the reason, afaik, Rutsey never told his version of the story and he did die recently at age 54 due to complications from his diabetes. It all worked out for the best that he quit/was kicked out of the band for everyone involved imo.
