The primary advantages to digital music are accessibility and portability. Compression software enhances both. You can fit your entire music collection on a single device and access it randomly. Those devices can be hooked up to networks or carried in your shirt pocket. Between a good Internet Service Provider and a laptop, you can take or access your music anywhere you go. The quality is often indistinguishable from the source material. While you can fit uncompressed music onto modern hard disks, it's hard to archive and transport that much data.
Compression software can shrink your entire family's music collection down to something that fits on a older hard disk. My music collection fits in less than 2 Gigs of hard drive space. Cheap raw music files take about 10 megabytes per minute of recording. Compression turns this into 2 megs/minute. My 600 songs take up 1.2 Gigabytes of hard drive space. In a world of 100 megabyte hard disks, this is nothing. You can put between 5 and 10 ordinary CDs on one CD, that's my wife's entire Beatles collection.
It's hard to overstate the advantages of having things accessible and portable. This script randomly picks out songs from my entire collection, from chant to Dash Rip Rock and plays them. There are many programs that do this more gracefully, but I'm too lazy to learn them. New storage media make it easy to carry much of that music collection with you. I can now fit my wife's entire Beatles collection on a compact flash (CF) card that works with my Zaurus.
The primary advantage of older media is durability. My mom's 40 year old records still play. CDs even have a nice life time expectancy. The problem is going to be finding a phonograph or CD player 20 years from now. Digital data, without Digital Rights Management, is easy to copy and move from one platform to another.
The problems facing Digital media seem even worse than dying phonographs but are not. "Where am I going to find an MP3 or WMF player in five years?" you might ask. The bad news is that you might not be able to find such a thing. The good news is that there are free file format alternatives that will be around.