MP3 is already a quite well known audio codec that is used mostly in mobile media players, which has now been dubbed as MP3 players, due to the significant reduction in file size it offers. AAC (Advanced Audio Coding) was a later addition to the MP4 standard and presents multiple and substantial improvements to the MP3 standard. Since both are lossy formats, it is therefore understood that they both sacrifice parts of the original audio data in order to come up with a much smaller file size. The key difference is in the quality of the recording after it is encoded with either AAC or MP3.
AAC has corrected a lot of the shortcomings of MP3 when encoding sound files and this has been duly noted both by technical people and casual listeners. The difference is so significant that it is not uncommon for people to equate the sound quality of an MP3 file coded with a specific bit rate with an AAC encoded file with a much lower bit rate. The difference in quality gets more pronounced as you go down to lower bit rates; 128kbps and lower.
The general advantage of the MP3 format is in its popularity. Because it was the first one to be introduced, it has become widely accepted by most software and hardware music players. AAC only gained popularity when Apple adapted it as the default format for its iPod music player and the music files sold in its music store. Before that, most portable music players, like those created by Creative, only supported MP3 files and not AAC. Now, most modern players being produced have added support for AAC and this gap is slowly beginning to close.
When choosing a file format for storing your music files, it is no longer under debate that AAC would be the clear winner. But before you encode your files to AAC, you should first examine if your existing devices are able to work with this format, as low quality is always better than not playable. When looking for a new player, its always better to find one with AAC support even when you have no intention of converting your files to AAC any time soon.
Summary:
1. MP3 and AAC are both lossy file formats for audio recording
2. AAC is generally better than MP3, more so at lower bitrates
3. The MP3 format has more support, especially in portable devices, compared to AAC