It's the same really except some people don't think it is right to say the name of God so they will say "gosh" instead.

Religious people don't "use God's name in vain" because they think it's blasphemous, or, more precisely, the Bible says that, not people. Thus, it was changed to "oh my gosh", but it does mean the same as "oh my God".

"Oh, my gosh" is a euphemism for "Oh, my god." In the U.S., talk is more frank then when I was young (in the 1950s) and euphemisms like this are old-fashioned. There are only a few really taboo words left. It is like "OMG." "OMG" is an abbreviation. But it is also a way to make the phrase more like a joke. Some other examples--all old-fashioned. "Jeepers creepers" (for "Jesus Christ.") "Judas priest" (for "Jesus Christ.") "Jeez" (for "Jesus.") "Heck" (for "hell"). "Darn" (for "damn.") "Tarnation" (for "damnation.") In the old Superman movies, Perry White, the newspaper editor is always saying "Great Caesar's ghost." "Caesar's ghost" is obviously another euphemism for "Jesus Christ" but I've never heard it except in Superman.

P.S. There are three kinds of bad language. "Profanity" means breaking the third commandment in the Jewish and Christian bibles, "You shall not take the name of the Lord your God in vain." "In vain" means not using the name seriously. "Oh my god, I'm late for work" would be an example of profanity. Profanity is not considered very serious in the U.S. these days. "Vulgarity" means bad language related to bathroom stuff. "Obscenity" means bad language related to sexuality. On National Public Radio, on a favorite program of mine, a humorous "news quiz," the host and panel use profanity, and they use some kinds of vulgarity, but they do not use any obscenity.