What is the difference between apply and put and smear?
Feel free to just provide example sentences.

"Apply" has a lot of meanings, but when it's similar to "put", the difference is that it means "put on for a specific use". For example, "apply paint to the wall (in order to color it)". It also sounds a little more formal than "put" in that usage. You would see it in instruction manuals, I think, but it's not too formal that you can't use it in everyday speech.

"Smear" is a very specific word that means you put something onto something, then messily spread it out? Sorry, it's hard to explain. For example, you can "smear paint on a wall". You can also have "smeared lipstick".