In the United States, they are pretty much the same "motorcycle" is much more common than "motorbike," but "motorbike" is used and understood.
For short, motorcyclists may be called "bikers." (And to keep the difference clear, increasingly people who ride bicycles are calling themselves "bicyclists.")
"Motorbike" is informal, almost a nickname. Sometimes, "motorbike" is used to suggest a smaller, lighter vehicle, but there isn't any precise distinction.
In the 1964 hit, "Little Honda," the Beach Boys sing:
"It's not a big motorcycle, just a groovy little motorbike;
It's more fun than a barrel of monkeys, that two-wheel bike;
We'll ride on out of the town, to any place I know you like."