What is correct, between
(a) Ich möchte das gerne machen.
(b) Ich würde das gerne machen.
or both? If both are correct, is there a difference in meaning?
Both are correct but there indeed may be a difference in meaning.
Who wants to answer the question?
Ich möchte das gerne machen. Wenn ich darf.
Ich würde das gerne machen. Ich habe aber keine Zeit.
Ich würde das gerne machen. Wenn ich darf.
The point is, when using möchte you say you're really willing to do something but you're waiting for permission.
Using würde can mean the same thing but it can also indicate that you would do it but you simply can't.
It's usually clear from context, however in case of "I can't" you tell why.
Adding to EM1's answer:
Ich möchte das gerne machen
can also mean that you just didn't get to do it yet, but are definitely willing to.
Eines Tages möchte ich noch an den Nordpol.
It is your wish to go there, but so far you didn't get to do it.
In addition to EM1's answer, which is 100% correct, I think if you take the standalone sentence
Ich würde das gerne machen.
this will be a little bit more polite – but this is a very small difference.
Also, this does not mean the other statement is impolite in any way.