TAW and AI offer completely different things in my mind. I don't think of them as remotely similar and therefore I don't see them as in direct competition.
There's only so many hours in the day though, personally I don't really have the time nor inclination to listen to two Liverpool podcasts. I jibbed Anfield Index a long time ago, probably around the time TAW expanded into TAW Player and thus filled more of a void.
I'd love to say I stopped listening because of a certain horrible human being on there, and he certainly contributed, but I just didn't think the standard of contributors (or indeed audio) was very high at all. It was boring listening to the same opinions on players being recycled. Some could do no wrong; some could do no right. Be arsed with that. The criticism of Rodgers crossed the line on many, many occasions as well.
When it came down to it, I was basically listening to an hour podcast just to hear Gags' (very good, if at times with limited context applied) section on pressing stats. I'm sure it has improved since - it certainly seems to have massively expanded as well, which shows they're doing something right. Good luck to them, but it's not for me.