What is the difference between “obrigado” and “obrigadinho”?
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It depends on your tone 
Grammatically speaking, the only difference is that “Obrigadinho” or “Obrigadinha” is in the diminutive, which we often use for emphasis or to be extra affectionate, for example. So, you can say “Obrigadinho” or “Obrigadinha” simply to emphasize how thankful you feel or to say thanks in an extra nice way. But we often also use it very ironically or even snarkily, like when something doesn’t go the way we intended.
Filho: Pai, posso ir à festa? (Dad, can I go to the party?)
Pai: Não! (No!)
Filho: Obrigadinho, 'tá?! (Well, thank you very much, 'kay?!)
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I have also learned that the diminutive “zinho” can, as @Joseph describes, be used in a sarcastic way. Suppose we encounter someone who thinks they know a lot about health or medical things, but really don’t, we could say “Muito obrigado, doutorzinho”! 
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