How different are the following two phrases? ① I promise. ② I give you my word.?

“I give you my word.” This expression is a strong way to say I promise or I guarantee. Here’s an example where you might use the phrase “I give you my word.”

Imagine that you lied to your boyfriend or girlfriend and they found out about it, and they’re very angry at you. Well, in try to resolve the situation and make peace with your boyfriend or girlfriend, you might say, “I give you my word that this will never happen again.”

The difference is in words only because the meaning is identical.

There's not a lot of difference. However, "You have my word" could be construed to imply that you have previously (already) promised (given your word) and this statement is a reaffirmation of what was said before, whereas "I promise" implies that you are just now giving your word. "You have my word" could be taken as present tense too, and would then essentially be the same thing as "I promise".

This kind of depends on what the person considers a promise to mean, and what giving one's word means.  To me, my word is my bond, so it is a promise.  Some people do not think that there word has much value so giving their word is not a promise at all.