In other words, can I do something with a volatile variable that could not also be solved with a normal variable and the Interlocked class?
Yes, you can gain some performance by using a volatile variable instead of a lock.
Lock is a full memory barrier which can give you the same characteristics of a volatile variable as well as many others. As has already been said volatile just ensures that in multi-threaded scenarios if a CPU changes a value in its cache line, the other CPUs sees the value immediately but do not ensure any locking semantic at all.
The thing is lock is a lot more powerful than volatile and you should use volatile when you can to avoid unnecessary locks.