Audio files are stored on a hard disk. You have two options to access them:
A streaming audio player doesn't have a HD.
It even doesn’t know where the audio files are located.
It connect over the network to a server.
The server “knows” the audio files and send all the information to the player.
This works pretty much the same way as you surf the internet using your browser.
It is called streaming because you don’t download an audio file in the way one downloads a file.
On request by the client, an audio file is send to client by the server. As soon as there is a sufficient amount of data (the buffer) the client starts to play.
It doesn’t wait for the entire file to be downloaded and it doesn’t store the file either.
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.
Sounds pretty complex but protocols like UPnP/DLNA come to your rescue.
Today devices discover each other on the network with the same ease as your PC discovers a external HD on the USB.
Although it took a long time for streaming to become mature (Slimdevices started streaming in 2000), today almost any audio brand offers amps/receivers DLNA enabled.
If you want to give it a try and you have two PCs running Win7 (or higher), 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 what 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.
Check if the streamer support this because even today support for gapless might be lacking.
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 the streamer supports this.
On the PC you can simply choose a media player supporting this tag.
Likewise custom tags, streaming protocols don't support them.
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.
The text above was written in 2015.
Today streaming has a very different meaning.
You have a provider (free or paid) like Spotify, Qobuz, Tidal, Amazon, etc.
Basically a very huge library on the internet.
This has become very popular.
Of course there is an app for it.
Often streaming audio players do have these apps build-in.
As much as downloads took over the CD, streaming audio services took over downloads.