I'll try to avoid the things people said already and just add my own experiences:
Biggest negative of the rain:
- Restaurants. Every place will be more crowded then usual cause you have people who just want to find a place to sit down and seek shelter. If you had planned to eat and it starts raining, you're not gonna have a good experience trying to find a table and wait on line. (even though people come into sit, they inevitably want to get some food to pass the time.)
I try to eat at 'off hours' anyway but counter service restaurants get like double capacity during thunderstorms. Priority seating restaurants won't feel it too much but some people may try to check in a little early OR linger on their meals to avoid having to go outside.
On the plus side, outdoor vendors will have shorter lines IF they stay open at all.
Biggest positive of the rain:
-Finding places in the parks whose themes are enhanced by the rain. A lot of the "village" areas of World Showcase in Epcot and Fantasyland in MK are actually really cool when it's raining. You get that memorable "huddled together" feeling while being in a "quaint village" during a rain storm.
Granted stores and restaurants will be more crowded but if you aren't planning on eating or shopping then you can enjoy the atmosphere.
I was in England (WS) at night during a particularly bad thunderstorm. It lasted two hours and there was frequent lightning. But I was able to walk through the connecting stores and stay dry while also going 'outside' (under a covered area) to watch the storm. Even worked up the courage to cross the street, get some food, and come back under the covered area to eat it. A very memorable night cause it had such a cool vibe to it.
The Norway store (that you go into after Maelstrom) is also a cool place to be in if its not to crowded. As is the Liberty Tree Tavern and oh...
The Columbia Harbor House. The best thing about the CHH is that it faces the Haunted Mansion. Go upstairs to the top level (if they let you) during the rain at night (especially EMH) and leer at the Haunted Mansion. See if it leers back