In film photography circles, there's this implicit assumption that when you "grow up" as a photographer you'll move up to the larger film sizes--medium format (i.e. 120 film) and large format. By that benchmark, I guess I've never grown up. There are times I've been tempted to try medium format, and in fact last year I did for a short while. I picked up a 1950s vintage Mamiya 6 Automat folder camera. While I liked the look of the photographs from that camera, I couldn't get comfortable with the viewfinder, so I sold it on eBay. Actually, my very first camera back in 1969 was a medium format folder, the Kodak Duo 620. I didn't shoot it for long before moving to 35mm on an Argus rangefinder. Once I started shooting 35mm SLRs back in the 70s, I was hooked.
35mm film cameras allow you to get in to capture photos in tight places quickly. That's why they became standard equipment for photojournalists, conflict photographers, and street photographers. If you're into the reportage style of photography made famous by Henri Cartier-Bresson, then 35mm is for you.
Another advantage of 35mm is the cost of the film. I just checked on B&H, and a 36-exposure roll of Ilford HP5+ black & white film costs the exact same amount as a roll of the same film in 120 format, which will yield 12 exposures on average. True, if you're after higher resolution or that medium-format/large-format look, then you'll have to move up to those larger film formats. For me, there's a low-fi look to 35mm film that I just simply love.
Finally, there is the availability and cost of the used 35mm SLRs. These cameras are the best value going on the used film camera market. True, you can find less expensive medium format cameras, but the good ones still go for more than 35mm SLRs, and for the most part, they are significantly bigger than 35mm SLRs. And don't get me started on 35mm film rangefinders. Those prices have gone crazy. For me, it's 35mm film SLRs all the way. And if you can't tell from the photo above, I have a soft spot in my heart for Nikon cameras.