Are they basically interchangeable or do they have different meanings?
They are not always interchangeable. They can have the same meaning sometimes, but destin has a more positive aspect, while you would use sort to express a more negative meaning. You would notice the same difference in English when comparing destiny and fate.
Sort can also mean spell or curse, which is not the case of destin.
Also, in some idiomatic expressions, you will obviously have to use one and not the other, like in:
- tirer au sort
- faire un sort (à quelque chose)
- c'est le destin!
For this kind of question, I recommend this tool which try to place English and French terms in a map even if it somewhat difficult to get used to it. You'll see that sort has a wider set of meanings, mostly englobing the one of destin.
Where they are nearly substitutable, sort tend to gives an impression of randomness while destin suggest something more forecastable.
Destin is like destiny. Something that was written in a metal book by God and from which there is no way to escape (like death and tax).
Sort is more like a fate. Something that crossed you way, either by a malicious wizard or bad luck (mauvais sort) or by random (tiré au sort).
A moins que ce sort n'ait été dans votre destin, il ne va pas durer.