The main difference between the two is that you usually make a promise to someone else, but when you make a commitment to do something, you might not even tell anyone else about it. There are also other problems with the way you've worded both sentences. "Commit" isn't normally a habitual action-- so rather than the Simple Present tense, the Present Perfect would be more appropriate: I have committed to studying for at least an hour every day to improve my English. You can use either the gerund or the infinitive after the verb "promise"-- but the meaning changes depending on which one you use. You could make a promise to someone by saying it this way: I promise to study for at least an hour every day to improve my English. (important note: not "to studying") Or you could offer your opinion to someone and/or give them your advice by saying it this way. I promise that studying for at least an hour every day will improve your English. One final note worth including is that you could also use "to studying" as a prepositional phrase like this: A commitment to studying at least an hour every day will improve your English. Just remember that when you say "to study" that "to" is a particle of the infinitive verb, and when you say "to studying" that "to" is a preposition.

Promise means telling someone that you commit. Committing can be done without telling anyone.

I commit to studying at least an hour a day to improve my English. (‘Commit’ is a little more formal. They aren’t made to others) I promise to study (for) at least…. (‘for’ isn’t necessary here but sounds ok. Promises are made to others)

They're very similar, but slightly different. -To commit is to dedicate your time or self to something or to an action, and you won't break the commitment. -A promise is very similar, but when we say promise, we mostly use it to mean the action of making that promise or commitment. So you can say, I promise to commit to studying everyday. Or I promise to study everyday. Or I am committed to studying everyday. (Usually, commit means to do an action, like commit a crime, so it sounds awkward even though it's not technically wrong.)

They both mean the same thing. "Promise" is usually used personally between few people. "Commit" is less personal language.