Which is higher quality: 256kbps AAC or 320kbps MP3?

As far as I know you will hardly hear the difference and if you happen to manage AAC will probably do better. As a pragmatic rule of a thumb you should be choosing AAC above everything else nowadays, unless you have reasons to choose something else. Here are all the actual competitors benefits:

MP3 - plays everywhere, usually in hardware (a battery-friendly way)

Opus - provides the best (AFAIK) quality for bits ratio possible today, does fantastic on low bitrates, fully open and royalty-free, would be the absolute winner if only hardware vendors were supporting it. But they aren’t. So it is played in software and can be considered battery-hungry if compared to MP3 and AAC. Choose it if you care about the storage space but not about the battery and if you want to be a free software evangelist.

AAC (M4A) - beats MP3 (though not Opus) in quality at equal bitrates and plays in hardware on almost every mobile device of the iPhone/Android era. Saves disk space and battery and sounds great.

FLAC - provides absolute lossless AudioCD quality at relatively low bitrates (varying around 512 kbps), free and open, widely supported in software, also supported by some car/home audio systems as an alternative to AudioCDs.

Vorbis can be considered deprecated nowadays. It still is beats MP3 but looses to Opus. APE used to be popular among audiophiles before FLAC has taken over.

256 AAC is equivalent, if not slightly better than 320 MP3. The high-frequency encoding is more efficient (more faithful to the original audio). Bear in mind that the ISO MPEG 1 Layer III (MP3) codec is 2 generations older than AAC.

320kbps VBR AAC is better than 320kbps CBR or VBR MP3.

AAC can store higher frequencies at the same file size as MP3, because it has better compression algorithm + it’s not outdated like MP3.

MP3 320kbps CBR cuts off at 20kHz while AAC is going above 20kHz, so you get better sound quality out of AAC which is closer to what FLAC sounds like.

AAC VBR 320kbps - 9.69 MB

MP3 CBR 320kbps - 9.64 MB

Actually, 256kbps AAC has a very similar sound quality to 320kbps MP3, although the AAC tends to sound slightly better than MP3, as you may already know, AAC is the successor of MP3 and it stands for: Advanced Audio Codec. If you encode your music to 320kbps AAC you’ll have a noticeable better sound quality compared to MP3 at the same bitrate

If you're going to be using it with anything other than audiophile grade hardware (the kind that usually costs upwards of US$500 and includes headphones and an external amplifier), just pick one - I think the differences would be minimal.

Everyday music devices are meant for music consumption, and not really for the pure enjoyment of exacting detail in terms of sound quality - and that's where the source matters a lot.

While AAC is supposed to be slightly better, the difference is not likely noticeable.  And the higher bit rate of the MP3 would likely make that a better choice.

AAC vs MP3 - Difference and Comparison