My take:
Despotism: good for beginners. Limits on food, trade, and production (mainly the first two) make it hard to grow massive cities, but you can maintain a huge standing army. I sometimes never switch on low difficulty levels and have no trouble winning quickly with a military victory.
Monarchy: Pretty useless unless you are at Prince or higher difficulty and have no other government options. Removes the food/trade/production penalties on your cities, so they will start to grow fast. Imposes a maintenance cost on each unit, but this is generally countered by the production improvement if you have a coal mine.
Republic: Very good benefit of increased trade. Otherwise feels like monarchy. Good if you want a high science society and race for the modern era techs without much conflict. Then switch to a more military government and crush their mustketeers with your tanks!
Communism: I don't remember why, but I always used to play this government on Prince or higher difficulty. It must have some kind of great benefit, but I don't recall. I almost always go for a military victory so it must be some kind of easing of penalties relating to military units.
Democracy: Less corruption, increased trade as Republic, but very very difficult to wage war. Good for the knowledge/production maximizing players who just want to have a spaceship victory.
If I'm playing a low difficulty level (Chieftain or Warlord) I often ignore government changes and play all the way through as a Despot. It's the only way to keep the game from getting rediculously easy, even after years away from the game. If I'm playing a high difficulty level (Prince or King) I usually race towards the Pyramids wonder and switch to Communism. Since I'm such a warmonger, Democracy was unplayably painful to me. I haven't figured out a good way to play the ultimate difficulty level (Emperor). I usually lose quickly, but I might have won once. I can't remember clearly.