"a photo of Mike" = Mike is in the picture. Mike is the main focus of the picture. "a photo of Mike's" = a photo that Mike owns, or a photo that Mike took. As Dan said, we don't know what is in the photo. There is a misleading answer to this question written by another commenter, so I want to clarify: there is nothing grammatically incorrect, unclear, or confusing about either one of these. They are both clear and correct. They simply have different meanings.

A photo of Mike is a picture of Mike. We don't know who took it or who owns it. A photo of Mike's belongs to Mike--and was probably taken by Mike. We don't know what is in the picture. It could be a photo of the Eiffel Tower or his cat, or a sunset in Oklahoma. One way to sum this up . . . IMO . . . OF can signify possessive, but in a case like this it doesn't, as that would be too confusing.

Don't say "a photo of Mike's". Say "one of Mike's photos", "a photo that belongs to Mike", "a photo that Mike took", "Mike's photo of his cat", "a photo that Mike has", or "Mike's photo".

The difference between "a photo of Mike" and "a photo of Mike's" lies in the possessive form. Let me break it down: "A photo of Mike": This phrase implies that the photo features or depicts Mike. It could be any photo where Mike is the subject. "A photo of Mike's": This phrase indicates ownership or possession. It means the photo belongs to Mike. The possessive form "Mike's" suggests that the photo is associated with or owned by Mike. In summary, the first phrase refers to a photo with Mike as the subject, while the second phrase refers to a photo owned or associated with Mike. The possessive form adds a possessive quality, indicating a connection between Mike and the photo.