Many Times vs Several Times
Many and several are synonyms. Because of that many times and several times are synonyms as well. They can be interchanged in any given sentence. There is only one slight difference between them.
Many times can be an undetermined number. This means that there is no clear number of how many times someone did something. Because it is undetermined you can use many times whenever you’re unsure about the number, or the number is big.
Several times means more than two, but not unlimited. You can use several times when the number is not undetermined, and not too big. Several times usually means up to six.