Do NOT name a tar.gz as .tar, it is annoying and it messes up some archive programs. There is no sense in it because it introduces ambiguity. You can have just a .tar that is not compressed.
You can use .tar.gz or .tgz as you like. The Slackware package manager make .tgz and .txz and will only accept these for install. Other than that, there is no reason to use one or the other. I prefer the shorter version .tgz.