What's the difference between 'remember studying' and 'remember to study'?

Use remember + ~ing to talk about the past.

Since 'remember' is the verb, you can use it in any tense. Present and future tense are most common.

I remember GETTING a new bike when I was in middle school. (I got a bike in the past. I am thinking about it now.)

I will always remember GETTINGt a new bike when I was in middle school. (I got a bike in the past. It was so exciting that I will think about it in the future.)

This man went mountain climbing with his friends. It was a difficult trip, but when he got to the top, he felt proud of himself. He will always remember CLIMBING that mountain.

Before she had a car, this woman had to walk everywhere. She will always remember BUYING that car because it changed her life.

I remember GETTING a new puppy when I was five. He was lying under a bush in the backyard when I got home from my friend's house.

Use 'remember to' talk about something you have to do.

Since 'remember' is the verb, you can use it in any tense.

I remembered TO PAY my electric bill. (I had to pay the bill, and I did it.)

I must remember TO PAY my electric bill. (The bill is due tomorrow. If it is late, I will have to pay a late fee.)

You are going to drive me to work tomorrow. Remember TO COME to my house by 9:00. Don't be late!

You must come.

I had homework for class. I remembered TO DO it, so my teacher did not get mad at me.

I had to do my homework, and I did it!

Your mother's birthday is tomorrow. Remember TO CALL her, or you will be in trouble.

You must call your mother on her birthday!