Agree with Nancy on this. The first phrase "it's an indescribable feeling" talks about a specific feeling in a given situation. While "it's impossible to describe a feeling" talks about general feelings being hard to describe. If you made the second phrase "It's impossible to describe THIS feeling" then the phrases mean almost the exact same thing.

Not quite. Change the 2nd sentence to "It's impossible to describe THIS feeling"; then they will match up.