PLENTY is an interesting word. It doesn't just mean a lot. Think of it as the opposite of NOT ENOUGH. So, most of the time, you'd use PLENTY if there was a specific purpose: He has a lot of land. He has plenty of land to build an addition to his house. She has a lot of money. If she wants to travel around the world, she has plenty of money. This is not a strict rule, but PLENTY is often use this way.

I think “a lot” usually would mean more than plenty. Example - there was plenty of room for everyone means that there was enough room for everyone. There was a lot of room means there was an excess of room.