As a verb, to INSTITUTE means to set something up, or to start something. To ESTABLISH means to make something solid or firm. Think of the word 'stable'. (Could also be to verify something). In normal conversation there can be much overlap between the words.

I agree there is not much difference. "Establish" means to make firm and "institute" means to begin something. I bet most native English speakers do not know the difference between the two.

There is not much difference. Tell us what you want to use it for, and that might argue for one word more than the other. You will see "establish" used more commonly and also used broadly.