Both are correct with different meanings. "Thanks, God" (with the comma) is a thank-you to God for something. A religious person may say it in a prayer to God, or a non-religious person may say it as a comment to themselves. For example, it has been very dry and no rain for months. A religious person may go to Church and say this as a prayer or personal comment to God. A non-religious person may simply say it to himself at home without any true religious intent. It would be simply a 'thanks' to 'whoever'. 'Thank God', most often is used for showing your relief that something difficult (in some manner) is over. Examples: "I had my exam this morning. Thank God, that's over !!". "I was on a diet for a month and was always hungry. Thank God the diet is over !".

Sorry, but the first one is not native English. The second one is; just note that we always capitalize all words that refer to the Creator of the universe.

"Thank God!" is correct. "Thanks God!" isn't exactly "wrong" grammatically, but it sounds like you know God personally. Natives don't say this, so you don't have to worry about it. There is a variant "Thanks be to God" but it's mostly only used by Catholics.