No. Hypnosis is not the same as being asleep. While the term hypnosis comes from the Greek word for sleep (hypnos) you are actually not asleep. You need to be able to listen for hypnosis to work because you need your brain to be processing words and responding to instructions while you relax and go into trance.
Some studies suggest that the hypnotic state is similar to the Rapid Eye Movement (REM) sleep state which is when dreaming occurs (whether you can remember dreams or not!). Others suggest that the hypnotic state is closer to that of Alpha when you have ‘in the zone’ levels of concentration and things just ‘flow’. Alpha is also the relaxed state you enter just before you drop off to sleep.
When you are hypnotized, you have an experience in parallel with your conscious awareness. You are fully aware of what is happening and where you are while at the same time you have an internally focused awareness.