You get to own a physical thing, with beautiful covers and sleeves.
New CDs are normally considerably cheaper than vinyl, and often cheaper than MP3 downloads.
Second-hand CDs are dirt cheap. You can buy thousands of them from Zoverstocks for a penny each. Take "The Queen Is Dead" as a quick example, on Amazon:
Seventy six pence! What else can you buy in the world for 76p that is as good as The Queen Is Dead?
They don't wear out.
They're harder to lose than MP3s. And if they do get stolen or destroyed by fire, you can replace them all through your home contents insurance.
You can give them to your grandchildren in your will.
They take up far less storage than vinyl.
They don't get warped when you accidentally leave them in the sun or next to a radiator.
You can buy a decent second hand CD player for much less than an equivalent quality turntable or MP3 player.
A CD player never needs adjusting. A CD player doesn't need its needle replacing.
You don't need to turn them over half way through an album.
You can buy them direct from the band at a gig.
You can easily convert them to MP3 or FLACs.
The majority of CDs now come in card sleeves which look attractive on your shelf (compared with plastic cases). Joanna Newsom's "Have One On Me" is a real thing of beauty.
You can get artists to sign CD sleeves. The recent Sun Kil Moon album came with a signed, limited edition live CD (not available on vinyl).
In terms of value for money, they're possible the greatest physical thing in the world. Small, cheap, high quality. Yet people constantly dismiss them.