What's the difference?
This one's fairly simple.
In British English, a cyclist is someone who rides a bicycle which is powered by their own legs by pushing on the pedals, or if there was some kind of motor, it would be an electric one fitted to a bike, and which can be charged up by someone pedalling the bike.
Now a biker can also referred to people who ride bicycles (particularly in American English), but the word, particularly in the UK, is more closely connected with people who ride around on motorbikes, usually in groups.
And that's it really...