‘Varied’ has a verb meaning. ‘Various’ doesn’t. (I varied the voltage to see what the effect was.) Most commonly ‘various’ mean ‘several’. (Various commentators have spoken out against the proposed law.) Not ‘varied’. ‘Varied’ is a higher register adjective meaning ‘containing a range of elements’. (The researchers theorized that eating a varied diet increases longevity.) Not ‘various’. You can express a similar idea in a more colloquial way by saying ‘they thought that eating a variety of foods might make people live longer. Another meaning of ‘varied’ is ‘containing variation’. I have a varied exercise routine depending on the season. = My exercise routine varies depending on the season. Not related to ‘various’. There might be some rarer meanings of ‘various’ with vaguely similar ideas to ‘varied’ in that they imply variety in a way that ‘several’ doesn’t, but not to the point where the words would be interchangeable.

I'm a native speaker, and I had to google it. I think generally they're interchangeable. If you google it, it appears there may be a slight difference. 'Various' is definitely more common.

Various is more common and has no reference to the nature of the things concerned - they could be different ie vary in any way. Varied, from the verb vary, implies that the things have been changed from their original state, so they are linked to each other. 'When we went through the documents we found invoices for various sums' - they may or may not relate to a particular transaction. 'When we went through the documents we found invoices for sums that varied' - there is an implication that these are all related to a particular transaction. The sums [of money] differ for some reason.

The only difference is the grammar. Varied is technically "past tense" (ie, "it has been varied"), while various is not specifically past or present or future, it's more generic. So, you can often swap these words without changing the meaning of the sentence. But sometimes it changes slightly. And sometimes one word just doesn't match the grammar of the sentence, so we need to use the other word. But aside from grammatical finesse, they have an identical meaning.