Yes, a DVD writer can also write CDs and CD-RWs.
The second ASUS drive you have linked can do the following:
Write Speeds:
DVD-R: 16X, 12X, 8X, 6X, 4X, 2X, 1X
DVD-RW: 6X, 4X, 2X, 1X
DVD+R: 16X, 12X, 8X, 6X, 4X, 2.4X
DVD+RW: 8X, 6X, 4X, 2.4X
DVD+R(DL): 6X, 4X, 2,4X
DVD-R (DL): 6X, 4X, 2X
CD-R: 40X, 32X, 16X, 10X, 4X
CD-RW: 24X, 20X, 16X, 10X, 4X
Read Speed:
DVD-ROM (Single): 16X max. (CAV)
DVD-ROM (Dual): 12X max. (CAV)
CD-ROM: 40X max. (CAV)
As far as purchasing goes, I would say go for a DVD writer (although you don't have to get a Dual Layer burner). Although I have a slot-loading DVD ROM (16x) a 24x CD-RW and an 8x DVD-/+RW drive, I only use the DVD writer on my desktop.
In a question of choosing between a DVD-ROM and a CD-ROM, a DVD-ROM is a better choice, just because they are not only cheap but it gives you the option to read DVDs.