HDD to SSD, not compatible

My dad has an older laptop that I intalled Linux Mint on a long time ago and he loves his Linux Mint. For Father's Day I got him an SSD but when I would get to the part of the installation where it needs to format the drive it would always give me an error and fail. A few times it did this after I swapped the drives out but now the computer will not even boot to a Linux Mint ISO USB or DVD with the SSD drive in because it gives an error saying that there's no hard drive detected. Upon further inspection I think I've found the problem and I think it's a hardware issue I just don't know what to do about it. The old HDD has 4 more pins off to one side than the SSD does so I don't think the SSD has a chance of plugging in, although I don't know why it was letting me get as far into the installation process as it did the first few times if it couldn't see the SSD anyway. Images attached, what can I do about this? Thanks.

HDDvsSSD.JPG