What are the advantages to a Class A amp & what are the trade offs?

Never buy based on design, etc. Buy based on how it sounds.

+1 to that by @millercarbon. I see many audiophiles who pay a lot of attention to the particular technology used in a product. To a large degree, the sound of a product comes from general implementation and engineering, not the class of operation, the chip used, and so on.

Also +1 to those who warned about heat. It is a real factor in choosing power amps, and Class A typically runs hottest.

To a large degree, the sound of a product comes from general implementation and engineering, not the class of operation, the chip used, and so on.

It’s a design choice. Ralph Karsten designs his Atma-sphere amps to run in Class A.
So does Tim de Paravicini. I believe it’s a design to lower distortion and colouration.
Nelson Pass designs some of the Pass amps and First Watt to operate exclusively in Class A.

I own an EAR Push-Pull Class A amp and the sonics are a  clean reproduction of the recorded material. The downside is the heat generated from the amp. But this was not a consideration when choosing which amp to buy, I chose the component based on performance and  
sound quality. 

People never complain of heat with tube amps only with SS class A amps. Yes, unless you get some trick Boulder or Krell design, class A amps run hot. The biggest problem with this is durability. Nothing benefits from recurrent heat cycling. A well built class AB amp will out live a class A amp built to the same spec. So, if you can find an AB amp that you really like, go for it. The problem is I can't. Not only that but power is everything. It is not unusual to have 12 db transients in music. That would be 16 times the power. So if you were cruising along at 8 watts and said transient comes along that will cost you 128 watts. If your amp can not support that you clip. Unfortunately for Boston Edison I am in love with big, heavy, powerful class A amplifiers. The best compromise may be the one that John Curl landed on which is build a big, heavy 400 watt class AB amp and bias it up to run 25 watts into 8 ohms class A. 
I wish I could tell you why class A amps sound better but I can not. Is it psychological? I do not think so. It is very hard to find a luke warm review of a class A amp. It might be because of the over built power supplies these amps usually come with that makes them sound more powerful than they actually are. There is also a certain airy effortlessness these amps seem to have that I have not heard in either AB or D amps. I favor the warmer tube like sound of Pass and Curl amplifiers. Some prefer the drier low distortion Boulder version. You either have to have a bunch of consistently good reviews,  try the amp in your system or a system like it before you pull the trigger. Of all the electronics in your system the amplifiers will make the biggest difference in sound quality.