There is no difference in meaning, as far as I know. "Mesmerize" is an out-of-date word for "hypnotize." The word "hypnotized" is used today, in a straightforward and literal way. Psychologists who study it prefer the term "hypnosis" to "hypnotism," and when it is used as a legitimate medical treatment "hypnotherapy" and "hypnotherapist" are preferred to "hypnotism" and "hypnotist." In any event, the actual, literal phenomenon is called "hypnotism." "Mesmerism" is named for a doctor. Let me check Wikipedia. It was named for Franz Mesmer, 1734-1815. It is an old name for hypnotism. It is not used literally any more. You might say "I am going to see a hypnotherapist to see if he can help me lose weight." You would never say "I am going to see a mesmerist..." "Hypnotize" is often used figuratively. "Mesmerize" is only used figuratively. In the figurative meaning, refer to something, such as a performance, or a natural wonder, that draws our attention so strongly that we forget where we are. We think of nothing but what we are seeing. "I was completely mesmerized by Cirque du Soleil." "I was completely hypnotized by the sunset and did not realize I had been watching it for half an hour, until I started to get cold." A person who has been hypnotized is said to be in a "trance," and you could call the sunset "entrancing." No, people do not talk about "hypnotizing" babies. A person who said they were "hypnotizing" their children would be regarded as strange. It would border on child abuse. So I am not sure what you mean here. If you gave a child a stuffed toy to take to bed at night, that had an electronic box that made soothing sounds--like a ticking, or a gentle breeze, or ocean noises--you would talk about "soothing" or "calming" the child.

They are actually synonyms of each other & can be used interchangeably (depending on what is being expressed) however hypnotize has a literal meaning of putting someone into a trance or a hypnotic state. Mesmerize just means to grab & hold someone's attention. Hope that helps! Not sure about "hypnotize the babies" maybe that's referring to soothing or calming? And you could use mesmerize to say you were fascinated by English language, but it would be more appropriate to say 'I was mesmerized by English'.