For a start, the second sentence is a past conditional sentence. The first sentence is not correct. It should say "Mike and his friends will go to see the races if it doesn't rain." The second sentence describes something that would have happened if a certain condition had not been fulfilled. The revised version of the first sentence describes something that will happen (in the future) if a condition is not fulfilled. In a future conditional sentence you should not be using the word "would."

Hi Daniel, In the first sentence, we are talking about a potential/future event. If it doesn't rain, they will go to the races. If it rains, they won't. In the second sentence, we are talking about an event in the past. They didn't go to the races because it rained. They would have gone, if it hadn't rained. Best wishes, Jon