Is there any difference between "I'm sat" and "I'm sitting"?
Solution 1:
Ignoring the grammarian 'it is wrong' response, the 'standard' (for want of a better term) answer is that it is a quasi-passive. Fowler, for example, explains it as such in his Pocket Modern English Usage. The basic idea is that sentences such as "someone broke the car" and "the car needs fixing" are passive-like in function though not form as the actor is external and/or unknown.
Maybe more relevantly, the same idea allows the passive voice to have a continuum of function for the past participle from adjective-like to verb-like. In this, sentences like "I'm sat/I'm stood" are more adjective-like in function while "I'm sitting/I'm standing" are obviously more verb-like.
In short, "I'm sat here" is similar to "I'm big" or "I'm tall" - you are describing yourself more than saying what activity you are engaged in. In contrast "I'm sitting here" is saying what you are doing.
In the spirit of being fair, I am not 100% convinced by this explanation but neither can I think of a better one.
Solution 2:
"I'm sat" assumes that 'sat' is the passive participle of 'sit', which requires that 'sit' be a transitive verb. (In AmE, 'set' is transitive, but 'sit' is not.)
Since "I'm sat" is passive, it means that someone put you in a sitting position, but that someone was not necessarily you, yourself; whereas "I'm sitting" means that you 'sat' yourself.
Solution 3:
I feel a little uncomfortable posting this as an answer, because it involves the death of a young man, and I don't mean to belittle his demise. However, from purely a linguistic point of view, the headline is of some interest.
The Mail Online, a British Newspaper tabloid, has the following sad title (Jul 10th 2016 2PM)
Pictured: Wife who was sat in the crowd as her Spanish matador husband was gored to death - the first bullfighter killed in 30 years
I'm pretty sure, ten or twenty years ago, the headline would have been
Pictured: Wife who was seated in the crowd ...
The construction, I believe, helps emphasise that the wife was one of hundreds (or thousands) of spectators in the bullfighting arena.