On the bike refers to the action of mounting the bike and riding it. 'I was on the bike when my phone rang'. 'By bike' refers to a bike as a means of transportation. Did you come by bike, or by car?

It means the same thing when you say "I came here on the bike" and "I came here by bike". The slight differences are that "on the bike" describes that you were riding on the bike, also the word "the" means that you're talking about a specific bike. "By bike" just means that you used a bike to get there, but doesn't specify which bike.

Thanks a lot. Is it the same with, for example, bus, pls? And when I want to say that I am on the roof of the bus, can I say just I am on the bus?

Yes, on the bike mean you are sitting on the bike. By the bike means you are near it, but you are not sitting on it. When I got on the bike for the first time, I knew I wanted to buy it. I was standing by my bike when a big dog came up to me. Hope this helps.