I'm wet with rain. - This means that the water on me is rain water. I'm wet from rain. - This means that I was outside when it was raining and my umbrella or raincoat did not offer enough protection to keep me dry.
I notice you use the past tense for the second one "I'm wet from rain". Would that implie "reason why you are wet" in the sentence? While "I'm wet with rain" having the focus more on rain water?