When there are several ways to say the same thing, it's hard for a native speaker to explain how they choose. The two phrases mean exactly the same thing. There might be a tiny difference in emphasis. "I'll put my warm coat on if it snows" puts a little emphasis on the choice of coat. One can imagine a longer version of the same sentence being "I'll put my warm coat on if it snows, and my spring jacket if it doesn't." "I'll put on my warm coat if it snows" puts a little emphasis on the decision to wear a coat. "Don't worry, Mom, I brought my winter things with me. I'll put on my warm coat if it snows."

There is no difference at all. Both sentences are correct and have exactly the same meaning. "Put on" is a separable phrasal verb, which means that you can say either "I'll put my coat on." or I'll put on my coat". NB Note that we don't have this flexibility when we're using a pronoun. The only option with 'it' is "put it on".

Both sentences mean the exact same thing. I would say the difference is in what the focal point is when you start off the sentence. In the first sentence, "I'll put my warm coat / on" you are talking about doing something to or with your coat. You put it "on" rather than in a closet, or on a hanger. In the second sentence, "I'll put on / my warm coat" you are starting the sentence focusing on you and what you are doing. you put on a coat, rather than a blanket or a hat. or even "I'll put.." the heater on, etc. Hope that helps!

If you're going to wear it then you should use "put on". The other sentence doesn't make much sense, it's totally unclear when you're going to put that coat in case of snow.