Oh, what a difference the final ‘s’ makes….  I hope this clarifies it for you.

‘Thank God’ (no ‘s’ at the end of thank) = if you are the one expressing your gratitude or relief about something, the correct expression (written or spoken) is ‘Thank God’. For example:

• “Thank God it’s Friday!”
• “I couldn’t turn in my essay last week because I was sick, thank God the professor gave me an extension!”
• “The school bus got into a minor accident this morning, thank God there was no injuries.”

‘Thanks God’ (with the final ‘s’) = here the word thanks has the final ‘s’ because it’s a verb in the Present Simple tense for the 3rd person singular subject. For example:

• My 85-year old aunt thanks God everyday that she can still walk on her own.
  [subject of the verb ‘thanks’ is ‘my… aunt‘ = 3rd person singular]
• He thanks God for each new day.

Wait… there’s one more!

“Thanks be to God” = this phrase is usually expressed in a formal religious context (i.e. a prayer/invocation), not in everyday conversation. Here, thanks is not a verb, but the plural noun form.