Why do certain fonts show up differently between the two above versions (same name on the drop down menu)?

Before I added version 2013 there were several fonts I purchased and/or downloaded on-line. Those fonts (possibly others as well) still show up properly when I use 2007, but look totally different when I use 2013. I can only assume there are some kind of compatibility difference between the two Word versions on certain fonts. I did some surfing and notice there are some "compatibility packs" that may help and can be downloaded. Will that solve the issue? What "pack" should I go with?

Appreciate any help.