Audio files are stored on a hard disk. You have two options to access them:
A PC has the audio files stored on a local HD and/or on a server somewhere in your home network.
A streaming audio player doesn't have a HD. It gets it music from a server. You connect to a server by providing its IP-address, browse its contents and choose a song.
Pretty much the same way as you surf the internet using your browser.
To play your own music using a streaming audio player:
After configuring everything correctly, the media player can access your collection over the home network.
A bit more complex compared with a PC.
If you want to give it a try and you have 2 PCs running Win7, enable media sharing and they can both play each other's content.
Modern TVs and smartphones are often DLNA enabled.
You might use your smartphone to play from or to a PC.
Gives you a taste of where streaming is about.
In general a PC offers more flexibility.
You can try all kind of media players, drivers, etc.
A streamer is a bundled hardware/software solution.
It does what is does and you don’t have much options to tailor it to your needs.
Multi room
This where a streamer excels. Just a tiny box instead of another PC.
However small laptops often rival streamers in price.
Sound quality
In both cases the sound quality is as good as the onboard audio allows.
In both cases you can use an external DAC to improve on it.
Streaming audio players often have a SPDIF (digital out) over coax or Toslink.
PCs in general don’t offer SPDIF but USB.
Gapless playback
In case of live albums or classical you need gapless playback.
If your streamer doesn’t support it you have a problem.
If your media player on your PC doesn’t support it you can switch to one supporting gapless playback.
Highres
Either you streamer supports it or not.
In case of a PC another sound card or an outboard USB DAC allows you to change this.
Tags
In case of classical you profit by having support for the composer tag.
Check if your streamer supports this.
On the PC you can simply choose a media player supporting this tag.
Likewise custom tags.
Power consumption
A small NAS and a streamer use less power than a PC.
A lightweight NAS might consume 15 Watt, a Squeezebox probably 5 Watt, a laptop probably 60 Watt
A typical audiophile worry is whether audio over the network will ever sound right.
Small wonder, if you pay Stealth Audio Cables $6,500.00 for a 1 m RCA interconnect to connect your CD player to your amp, you can't believe that sending audio over a CAT5 ( $50,00 for 100 meter) or over Wi-Fi can sound right.
The answer is simple, you don't send audio over the network, you transfer data from one device to another. Networks are built to do bit perfect data transmission.
What is sold as “music server” is in general a dedicated audio PC.
Most of them are not servers properly as you can control them directly and do local playback.
However, a lot of them can be used as a (DLNA) server as well.
They have the look and feel of traditional audio equipment. In fact some companies stress that it is NOT a computer. They probably think they fence people of by calling it what is, a computer. More can be found here