I prefer an MP3 player over my phone. Here is the one I use. Why I like this one:

• Dedicated device designed for music.
• Hardware designed to play high quality music. (Think using Ubuntu vs Ubuntu Studio for music production)
• Dedicated buttons instead of all touch screen.
• More options for integration with other devices or systems
• No distractions. Phones nowadays demand our attention for every little thing. Every app, no matter what it is, has notifications.
• The Bluetooth is better.
• You can literally hear the difference in the quality of the music if you use good quality headphones/ear buds. The same song, same file, will not sound the same if it's a good quality FLAC.

While I use my phone for music I can certainly see some advantages:
physical buttons
smaller and lighter
less distraction than a phone
cheaper to replace if stolen or broken

I have a shuffle just for running.

Having something that doesn't connect to the internet makes for a better device in the long run. Plus, as others have mentioned, less distractions.