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

Stop + ~ing answers the question 'what'.

This man has to stop PLAYING video games. (What does he stop? Playing video games.)

The stop can be temporary. (He will not play video games for an hour so that he can eat. He might play again after dinner.)

The stop can be permanent. (He will never play video games again!)

If you want to be healthier and save money, stop SMOKING.

I have a headache. I wish this child would stop CRYING!

This young woman ate too much, and now she feels sick. She should have stopped EATING sooner.

Stop to answers the question 'why'.

I am driving down the road and get thirsty. I need to stop TO GET something to drink. (Why do I stop? To get something to drink.)

We often say 'stop and' instead of 'stop to'. I need to stop and get something to drink.

If you are going on a long trip, stop TO GET gas before you leave time. Otherwise, you might run out.

These men don't know where they are, so they stopped TO CHECK the map.

This little boy was playing in the park. When he saw the beautiful flowers, he stopped TO SMELL them.

'Stop to smell the flowers' is a common expression in the US. It means that we have to slow down and appreciate life. When your life is crazy with work and school, take some time to stop to smell the flowers!