Apple's Voice Memos has been around since iOS 3, and in iOS 7 it got a new look and a new icon, but no new features. You can record a voice memo, trim it, and email or text it and Voice Memos also supports AirDrop.
It's a good audio recorder, but it's not for the power user. Audio Memos is far more powerful, and it's only US$0.99. Many users will probably end up buying all of the in-app purchases that take the app price to $9.99. If you can pick and choose to get the features you really need, Audio Memos is powerful, inexpensive, and useful.
Like Apple's Voice Memos, the basic Audio Memos app is, well ... basic. It records voice memos, and it supports iCloud, so you can grab your memo from any device signed into your iCloud account. Voice memos can also be emailed to recipients. The app records in the background, so you can record a lecture while checking your email or playing a game if you like. Apple's Voice Memo also does this. When in recording mode, a red bar is visible across the top of your screen. Tap it and you are returned to the Audio Memos. The app also supports selective erase of part of a recording, but no fancy editing or trimming. Even at the basic level of the app, there is a lot of power.