There are many small disadvantages to Quick Speed.
Some of them are:
-inefficient whipping
-multiplied lost of turn while moving workers and settlers to their destination
-5 turns without being able to change civics again (same in all speeds!)
-10 turns to dow again (same in all speeds!)
-10 turns to break a resources trade (same in all speeds!)
-50 turns to get +1 because of peace (same in all speeds!)
-Ditto for every relations modifier
-etc, etc, etc
The biggest advantage of Quick is the one mentioned by Harbourboy, and thats the reason I play it in my quest for the perfect strategy.

To the admins:
Could the dates mentioned by Lexad, the 25000 instead of 33333 mentioned by WatinTime and the small details in this post be fixed in a MOD?
I really hate this out-game factors. I dont want to talk about rounding mistakes in years in different speeds, I want to talk about different strategies when playing the game
Which takes me to FiveAces proposed strategy. I like it, it is simple and flawless. Just try it and win the gauntlet! If you dont win, well, just propose another strategy. That's the way to progress. Just a comment, why delay the building of the third city? Build it soon, get missionaries out of it, cottage it now that you have it
I'm considering it partly for you, and partly because I really because of BTS. I'm not sure I can come up with any other good Vanilla/Warlords Majors when the new XP is drawing in people, and we seem to have a warm reception to this one.
Oh, if that's the reason, then please, go for it!!! Pretty please!