You shouldn't try to learn another language based on translations. Your are not automatic translation software, so you'd better try to understand the underlying meaning of the original words instead. So let's see what "mucho" and "gusto" mean:
Mucho: "Numeroso, abundante, intenso."
Gusto: "Placer o deleite."
So we have "mucho" equals "a great amount" and "gusto" equals "pleasure". So "mucho gusto" equals "un gran placer" or, as you say in English, "a great pleasure".
Answering your question, there's no difference between "mucho" and "muchas" beyond the use of the masculine/feminine and singular/plural to match the noun it affects ("gusto" or "gracias"). In both cases it means "a great amount" or "abundant" or "above the standard amount":
"Muchas gracias": My gratitude is above the standard amount.
"Mucho gusto": My pleasure to meet you is above the standard amount.
Your confusion with the translation of "mucho gusto" as "nice to meet you" is that the word "mucho" is not translated. You have translated "gusto en conocerte" instead. Maybe a more accurate but not idiomatic translation would have been "It's very nice [to meet you]", so there you have the word "very" that maybe helps you with understanding the word "mucho".