I hated Twister for the plot, the stupid characters, and ridiculous events. I loved Independence Day for the same reasons. :-) While both are disaster movies, there are clear differences between them: Primarily, Twister took itself far too seriously with an overly dramatic (divorce) relationship, and the good vs. evil characters (the competitive Tornado Chasers), and all that. ID4 had a small amount of that, but had its tongue firmly in its proverbial cheek.
The plot was simple and straightforward: aliens come to destroy the earth. There's nothing to believe or not believe about this. It's not a realistic (as far as we know) event, unlike tornados, which are more down-to-earth, if you'll pardon the expression. :-) So, right off the bat, we are given a premise which is already over-the-top. Combined with the huge, over-the-top space ships and other special effects, we are poised for sensory-overload, which is a different entertainment effect than what Twister provided.
The comic relief was effective was at the most important level: the premise. That is, they incorporated the "Area 51" controversy into the movie. This is a Tabloid-favorite, along side Elvis sightings, as one of the most popular stories to write about in National Enquirer and Weekly World News. It's a nutty theme that no one really takes seriously, almost to the point where not even The Bundy's (from Married with Children) would believe. That is, the government has been horboring aliens that crash-landed here since the 1950s. And just like with the Elvis sightings, where the tabloids print pictures of elvis in different contexts, we have seen similar photos that were purportedly "snuck out" of the base there and these newspapers have exclusive photos. These things are just as entertaining to play with and it's more fun when everyone participates! By portraying the Area 51 theme as a realistic event, it's almost celebrating the silliness of the whole thing.
In the same light, but a little more subtly, I thought it was great that they used Brent Spiner (Star Trek's "Data" android) to play the scientist from Area 51. The satire cannot be more clear: he's normally this logical android, but now acting like a complete loon.
ID4 isn't a great movie. It's just a fun movie. As someone once said,
"it's just good clean fun for the whole family." In summary, it was a
better-than-average movie because of the special effects and a theme we
haven't seen in the movies for years.