As a kid I was always told off by my parents for saying 'different to' instead of 'different from', but I was never really tempted by 'different than', which has always grated a bit. Here's my ham-fisted attempt to explain why:
'Than' is usually used to introduce the second element in a comparison (according to Oxford): 'She is smarter than me'. In other words it's used with a marker of comparison ('more', 'less', '-er') along with an adjective or an adverb. 'Than' doesn’t really work without that marker. The sentence 'She is smarter than me' should probably be read as 'She is smart [er than] me', with the [er than] constituting more of a 'word' than 'than'.
'Different' isn't a comparative, and doesn't have a marker of comparison. It kind of leaves 'than' incomplete, bereft of it's better half.
The other main use of 'than' is in combination with words like rather
('I'd rather play at Wimbledon than watch it on TV') or other
('I was unable to do anything other than fall to my knees and weep'), but 'rather' was originally a comparative (of the adjective 'rathe', according to Oxford, meaning 'blooming early in the year'
), and I wouldn’t be surprised if 'other' was too (since it ends in 'er').
Another way to look at it (if anyone’s still awake...) is to think of maths: '<' means 'greater than', '>' means less than. Different is just '=' with a line through it.