The issue is that AVCHD's are burned onto DVDs but they are encoded in Bluray format. Your computer can read an AVCHD even if you only have a DVD drive/burner in it because the programs you have installed on it can recognize the file format/structure. That is to say, you can watch the movie when it is on your hard drive, so the fact that you're accessing the data from a different location (that location being a DVD disc) makes no difference to your computer.

A standalone set-top DVD player on the other hand (ie. devices that use the technology that was available before Blurays were invented) is looking for a specific standard DVD file/folder structure and doesn't understand what to do with .m2ts files and such. It doesn't just have to do with the physical disc; it has to do with the way the files on the disc are organized/named, and how they are encoded.

Suffice it to say that the software of a standalone DVD player doesn't have the proper codecs installed, while your computer does.