To some extent the answer depends upon exactly what you're going to burn onto the CD-R. If you're going to burn copies of a vinyl collection, then a stand alone recorder is the easiest way to go. For $400-600 at your local music chain (Sam Ash, Guitar Center, etc.) you can get an excellent sounding, simple to use stand alone recorder. The stand alone models will also do a sonically excellent job of copying CDs, but it does have some drawbacks. Making copies using a stand alone player is very time consuming in that it's done in real time.
If you use a computer with a built-in CD-R the process is much, much quicker and affords you several valuable convenience features. If you use Apple's free software, iTunes, you can rip a CD in 5 minutes onto your hard disk. Assuming your computer has an internet connection iTunes will automatically supply the artist name, album title and individual song titles as part of the ripping process. You can then also use iTunes to burn a copy of the just ripped CD. The burn speed is user variable, but I can't honestly say I hear any differences between the burn speeds, hence I use the highest possible and the burn process takes 10 minutes. You can then use a print screen utility to print a list of the songs you just burned and use it as the CD insert.
Another big feature of using a computer is that once you've ripped a number of CDs onto your computer hard drive it's now just a click and drag operation to make compilation CDs. You can make a compilation of your favorite 17th century Ukrainian folk tunes in less than 15 minutes. Making the same compilation using a stand alone recorder would as a minimum take an hour. Don't underestimate iTunes ability to quickly access songs once they've been ripped, it's an enormous convenience feature.
Computer based systems can also record vinyl to digital, and ultimately they can perform a far more thorough and better sounding conversion process than a stand alone recorder, but you would then require an expensive sound card and/or expensive outboard A/D converter plus the appropriate recording software. We're talking in excess of $1,500 plus the cost of a computer. The big advantage of such a system is that it will allow you to digitally process you music. This can result in a substantial reduction in clicks/pops and other vinyl surface noise. Also, a little EQ can do wonders to what are otherwise sonically annoying albums.
My bottom line recommendation - get a new Mac computer. When compared to a stand alone recorder it's quicker to use, is far more flexible and can be further upgraded for increased functionality. A stand alone recorder is your best bet if price is serious factor and your aim is to make casual transfers of vinyl. A basic Mac with a built-in CD-R costs as little as $799 (including monitor). A new G5 with built-in digital optical I/O is priced at $1,799 (plus monitor). Used Macs are readily available on eBay.