I would use them in the same way for meaning, to introduce an important thought or bit of information. But 'the thing is' can't be a sentence on its own. - The thing is, your cooking is terrible. - Here's the thing. Your cooking is terrible.

I don't think there's any difference at all. Be aware that both phrases are informal. They are not slang. They would be used in conversation, even by highly educated speakers. They would be used in writing that has a "conversational" tone, such as a magazine article. However, they would not be used in formal writing, like an academic paper, or a typical piece of "expository prose."