The Guitar rig stuff is basically just software you install on the computer. I don't play live, and just record by plugging my guitar directly into the audio interface, and use guitar rig to simulate an amplifier/cabinet/gizmo.
The Line 6 POD HD500 Multi-Effects Processor is more of a live performance tool, in that you put that on the floor, plug your guitar into that, then into the amp.
Kind of apples and oranges. Both have their place.
To add to this, you can use the HD500 (which is the newer version of my POD XT Live) into headphones or a computer (for recording), but it really shines for live performances. One advantage that the HD500 also offers (even for playing at home) is that you can use your feet to change your sound in the middle of a song. With computer-based modeling, you typically have to use a mouse to change patches. For some people, this is a big deal, and for others it isn't.
I suspect that the Guitar Rig software has at least as many amp, speaker, and effect models as the POD, but I'd also suspect that the POD patches are a bit more authentic-sounding. (Line 6 has been on the leading edge of modeling since the beginning. They're sunk a ton of money into research.) But keep in mind McMillan's Theorem: Anything with more than three knobs on it has more bad settings than good ones. Sometimes simple is better. Creating a good sound with so many choices (with either platform) can be a challenge.
One thing I appreciate about the Line 6 stuff is that they have a large user base and a lot of people are sharing different patches. (All the settings can be saved on a computer and shared with others.) For example, I needed a good set of patches for "Blue on Black" by Kenny Wayne Shepherd, and it took me all of about 3 minutes to find several, download them, and try them on my POD. Same thing with patches for "Comfortably Numb" and several other songs we play. Instead of spending hours dialing in the right sound, I just find one that some 14 year old kid with too much time on his hands has posted on the Line 6 forums.
I'm not familiar with Guitar Rig, so I don't know if there's a similar user community.
I have also learned that the POD (and its brethren) sounds best running into a full-range amp like a keyboard amp or PA system (or headphones, if you're just playing at home). It can be run into a regular guitar amp, but since a guitar amp typically adds its own coloring to the guitar's tone, it ends up defeating some of the coloration that the POD is providing. Up until our last gig, I was running the POD onstage into a Behringer keyboard amp as a guitar monitor pointed at me, and running a line out to the PA system. We found that my guitar was too loud on stage (making it hard for Mike to hear his acoustic guitar, which is also running through the PA) and not loud enough in the audience. So the last gig I did away with the keyboard amp completely and ran the POD directly into the PA. Then we increased the volume of my channel on the PA compared to what we'd been running before. The end result was much, much better. Since I typically have a PA speaker directly behind my head, I had no trouble hearing myself. Mike could also hear his acoustic guitar clearly through his wedge monitor, and because of the increase in my channel volume, the audience could hear me better than ever. As a bonus, it's one less heavy thing for me to carry to and from the car.