Generally, obliged is a feeling of being in debt to repay a favour, while obligated is a legal or moral requirement to do something.
However, some published definitions do allow for each word to be treated as synonymous, with both meanings attributed to each.
It’s common to hear the words obliged and obligated used interchangeably, suggesting that they mean the same thing. In fact, there are slightly different meanings attributed to each
According to Grammarist – to be obliged is to “be in someone’s debt because of a favor or service”. While the word obligated “carries a slightly different meaning, which is to force someone … to do something because the law or morality requires it.”
As a verb, obliged has a similar meaning to obligated but without the legal or moral connotations. Instead, it connotes an action done with no expectation of a return favor.
To put that a different way you can feel obliged but not obligated, and you can literally be obligated, but not obliged. For example…
“Because you’ve been a good tenant I feel obliged to tell you that, under your rental agreement, you are obligated to pay your rent on time every month”
With that said, there are numerous sources (Merriam Webster, grammar.com) that indicate that the word obliged can also relate to a legal requirement, rendering the two somewhat synonymous. It seems that fewer sources (e.g. Longman Dictionary) attribute the feeling of needing to repay a favour to the word obligated.
So, you’re not wrong if you use each word for both a feeling of moral responsibility and a legal requirement, but I for one, prefer the clear differences between the two and plan to guide my speech accordingly.