There is no difference in meaning between these three words, but there are some slight differences in the way they fit in the sentence.
Help is the most common and most informal (aid and assist are both more formal). Aid is more commonly used as a noun, not a verb:
There is also a word aide (pronounced the same way!) which refers to a person whose task is to help – an assistant. A nurse’s aide, a teacher’s aide, etc.
Now let’s look at the two verbs: help and assist.
After help, we can use a verb with or without “to”:
In everyday spoken English, it’s probably more common not to use “to.”
After assist, we must use in + -ING form of the verb or with + noun:
I’m happy to assist you with your website.
This program assists people with their job search.
The noun form of help is also help, but the noun form of assist is assistance:
Thank you for your assist.