What is the difference in a single ended solid state class A amp and one that is not single ended?
A single-ended class A transistor amplifier will be very inefficient. It will also be higher distortion, as the 2nd transistor of a push-pull class A design will help cancel certain distortions, in particular even-ordered harmonics.
This is true of SET vs push-pull tube amps as well. However I would expect an SET to manage better than a SE transistor amp as triodes are inherently more linear. Note also that coupling a single-ended transistor amplifier to the speaker has some challenges which are likely best met by the use of an output transformer. It is for this reason that single-ended transistor amplifiers will be extremely rare.
One thing that has not been mentioned about push-pull and class A is that its not entirely about crossover distortion! If you read between the lines my explanation above points to it- by having two output devices track together in opposition, you have distortion cancellation all the way up to the full power of the amp. In a class AB situation this is not true- so they tend to have more distortion at higher power levels (outside of the 'A' region). This can cause the amp to sound good at low volumes but get harsher at higher volumes. Many of you may think that all amps do that but you would be mistaken :)
Now here are some other facts about class A:
It can be used single-ended or push-pull. It can also be tube or solid state. Because class A is more linear, it also makes possible the ability to run the amp without feedback. This is true whether single-ended or push-pull, and/or whether tube or transistor.
(In the world of tubes there are also two variants of class A, A2 and recently an A3 design has been introduced by Electra-Print of Las Vegas. Neither A2 nor A3 can be executed in a semiconductor embodiment as the output devices would be destroyed in the process, but tubes can and do allow this operation with good reliability. A2 and A3 describe amplifiers wherein the output devices do conduct during 100% of the waveform at full power, but have operating perimeters set up in such a way that considerably more power is available, although there may be considerably greater requirements to drive the output tubes with linearity. Fisher made a class A2 amplifier called the A-50 back the 1950s, Atma-Sphere makes Class A2 OTLs right now and Electra-Print of Las Vegas offers a class A3 SET amplifier.)
Since feedback is associated with odd ordered harmonics (which the ear finds quite unpleasant), there have been a lot of designs showing up in the last 10-15 years that are class A/zero feedback in an effort to make the amplifier sound more like real music. This is the impetus behind the SET movement, Nelson Pass' '1st Watt' amplifiers, the Ayre amplifier, certain triode push-pull amplifiers (VAC makes some if memory serves) and our own amplifiers (OTLs).
However class A is a tool in a designer's kit, so to speak, and like many other aspects of amplifier design, are not the make or break indication of the sound of the amplifier. However, it *is* usually a good sign :) that things will go well.