What is the difference between "I have got to study on weekends" and "I have to study on weekends"?
Have to X means to be obligated to do X or to be required to do X.
Have got to X is an emphasized version of this. Usually when this is said, X or the condition requiring X is urgent or happening soon, or been around awhile and requiring attention right now. It's informal/colloquial.
I'm getting tired and hungry. We have to get out of here soon.
That building's on fire. We have got to get out of here right now.
Going with your examples:
I have to study on weekends.
Something is requiring you to study on weekends. Perhaps a family member is insisting you spend your free weekend time studying, or it's the only free time you have.
I have got to study on weekends.
This sounds like you have a test coming up very soon and really need to study all you can now, or you failed a test because you weren't studying and are telling someone who is disappointed in you what you can to prevent an issue in the future.