I for one don't have the patience for correspondence chess and it is reflected in my rating. My ratings are mostly equal across-the-board. In my opinion and this is only my opinion, people who have a Blitz rating of 1200 and a correspondence rating of 2000 are most likely cheating on the correspondence. A rating should not very that much. There are so many ways to cheat (have a friend who's higher-rated play for you, ask a higher-rated friend to help you on the moves, chess engines, etc.) and so many different reasons why to do it. (Bragging rights, trying to feel important, wanting to be noticed, etc. etc.)
As far as how important Internet rating is compared to regular rating I think that's all depends on the person. I don't have very many if any tournaments for adults being played in my area, so the Internet is pretty much all I got. I have to agree with Kasporov_Jr mostly on this one...
it's true bro, cheating in correspondence chess is so easy & hard caught, you can literally use a chess engine for 2-3 moves a time multiple times during a chess match without looking obvious.
in blitz/bullet, chess knowledge is all raw power, eithe you know it or you dont. thats why I respect people with high blitz/bullet than CC . And people with extremely high CC ratings, but low bullet/blitz is extremely sketchy. Pretty much every modern day GM like Carsen, Nikamura, Anand are all extremely good at blitz/bullet as well.
Like other people have said, it's not any harder to cheat in blitz/bullet than it is in correspondence.