Alnico has at least two desirable properties:
First, alnico is less subject to flux modulation under dynamic conditions than are ceramic magnets, and this has been shown to be audibly significant (as described by Dr. Earl Geddes in "Transducers"). So all else being equal, an alnico magnet woofer will be more linear than a ceramic one within its safe operating range.
Second, alnico is electrically conductive, which effectively lowers the voice coil inductance via shorting (voice coil inductance is a primary limiting factor of a woofer's high frequency extension).
One drawback of alnico is that if the magnet is overheated, it will permanently lose flux (and have to be re-magnetized to restore performance). If a ceramic magnet is overheated, it will lose flux temporarily - that is, the flux will return once the magnet cools down. This was not an issue back in the 50's when amplifier wattages were low, but can be a factor today.
A ceramic magnet woofer with a well-designed, well-cooled magnet structure and a Faraday ring (copper or aluminum shorting ring) is competitive with an alnico unit, and doesn't have the susceptibility to permanent demagnetization if overdriven that the alnico has. The manufacturing cost of such a woofer is usually less than an alnico one, but is still high.
As an example, Pioneer makes some excellent alnico magnet prosound woofers, typically rated at 300 watts. This is more than enough power for home or studio monitor use, but probably not enough for dance club or sound reinforcement applications. A number of manufacturers make ceramic magnet prosound woofers that can handle 2000 watts or more.
Alnico magnet drivers are often described has sounding more lively and/or more harmonically rich than ceramic magnet drivers, but unless the comparision is with a very high quality ceramic unit I'm not sure it's a fair comparison. I don't have a hard and fast opinion on which is best for home applications; I've heard excellent speakers that use both types of magnets.
I think that the high quality woofer magnet material of the future may well be neodymium, which from what I understand has many of the desirable properties of alnico but is more rugged thermally (still not as bullet-proof as a ceramic magnet, though). Cost is in between alnico and ceramics, but neodymium magnets can be made much smaller and lighter than comparable alnico and ceramic magnets which can reduce manufacturing and shipping costs.