I suppose for the most part, the big difference is in the Input/output options. I use a 'proper' external soundcard and good headphones for music, but for 90% if purposes it dosen't matter. I've an unpublished article about the three main options for sound cards, and this is taken from there.
Firstly, the traditional sound card and your integrated sound devices.Integrated sound devices usually are designed around providing the bare minimum of outputs at the lowest cost. While there’s some motherboards with fancy features, in most cases you’d have 5 channel sound and SPDIF. While soundcards were similar, since basic features have been commodified, these often offer better headphone amplification stages, more options, and ostensibly better sound chips. If you’re a gamer, soundcards are likely the best option, since they’re built around gaming needs, such as surround sound
The second is the DAC or digital audio converter. These are units used by audiophiles which usually do purely output and either have a headphone out, or can be connected to an amplifier. These tend to be somewhat expensive, you’ll need to do your research but in many cases should offer the best sound quality for music. These are usually connected to your PC via USB and are detected as a soundcard. These tend to be expensive (though prices have been dropping), and as with any audiophile grade equipment, you can’t always trust reviews. Do lots of research, and check out as many reviews you can.
The third is the DAW or digital audio workstation - these are basically DAC-type devices with better inputs - I use one of these for playback, and if you do your homework, it offers good sound quality, often at lower prices. They’re pretty good if you want to use proper mics (with RCA/Phantom Power), or want something that can output to RCA - my music playing setup is built around a relatively inexpensive, rather old M Audio fastrack, so I obviously believe this is a good option.
If you have specific needs, a discrete sound card can be a very good idea. For most people, less so. If you do, you're probably going to have to do some legwork to find the right soundcard, often based on the choice of amplification (if any) and output you'll need. You can find some with kickass headphone amplification, or use RCA for an external amplifer. If you need to ask if you need one, chances are no, modern soundcards are good enough.