Other than that they are both white, I think there are far more differences, than similairities. Different flavor profiles in most cases.
I'm not a fan of most domestic Rieslings, but love a ton of well-made GR and FR versions. For Pinot Gris/Grigio, I find that some of the OR/WA versions are better, than all but the better FR versions. I have yet to find an IT version, that bowled me over, but then this is a varietal, that I do not actively seek out.
As Robert suggests, stop by your local wine shop and grab versions from the Alsace. Keep them similar in price range, so it's less apples to oranges (or Rieslings to Pinot Gris), and sample them.
Here I go with a generalization, that will cause heat, but do not attempt to compare "inexpensive," versions of say CA Riesling vs IT Pinot Grigio. Who knows, with these wines. Get the "good stuff," and try to keep the geography and the price-points close, to do a side-by-side comparison.
To me, well-made Rieslings are about as food friendly, as a wine gets. This includes a lot of GR and the Alsace Region of FR. Pinot Gris/Grigio, from the Pacific Northwest, goes well with salmon (when it is well-made). My favorite is still the King Estates Reserve (note: the Reserve, and not their "regular" release) PG, and it goes well with Alaskan salmon. Same for the PN from that region.
If I were doing a taste-test of the Alsace, I'd also throw in a good Pinot Blanc, as well, just to confuse things.
Given a similar geographic region, similar price-points, I think you'll be looking at two different grapes, and also two different styles of wine. One could easily set up a fun evening.
Both Riesling and Pinot Gris can be produced in a wide range of styles. Both can be made very dry, or they can be very sweet (e.g.: Botritis-affected, syrupy sweet dessert wines). Pinot Gris, in particular, can also vary widely in terms of body -- think light-bodied Pinot Grigio from Italy's Alto-Adige versus a full-bodied Pinot Gris from France's Alsace region. (Personally, I find the former ranges from OK to quite good, and generally suitable for quaffing on a summer's afternoon, while the latter is more rich, more spicy, more flavorful and more suited toward the evening meal.)
Riesling will *generally* be higher in acidity than Pinot Gris, which -- even as a bone-dry wine -- will be rounder in body and fleshier in texture. (Indeed, most of this post is filled with generalizations, and there are exceptions to almost everything!) Rieslings will generally mature longer/better with bottle age than will Pinot Gris.
The best way to understand this is to do as Robert suggested. Ideally, you would grab both a Riesling and a Pinot Gris from the same Alsatian producer (e.g. Trimbach, Hugel, Schumberger, etc.), or from the same California producer (like Navarro). That difference -- as long as it's the same quality level (you don't want to compare a Pinot Gris SGN with a Riesling Reserve Personnelle) -- will maintain throughout each range/quality level.
Once you've done that, compare a Riesling (or Pinot Gris) from one location to a Riesling (or Pinot Gris) from another -- say, a Riesling from Alsace with one from the Mosel; or one from the Rheingau with one from Australia; etc., etc. Try an Alsatian Pint Gris next to an Italian Pinot Grigio, or one from Monterey Co., Mendocino Co., or Oregon . . .
Most California Riesling is not very good, Pinot Grigio ranges from pretty decent to weird, so with most wineries that might be a misleading comparison.
The Navarro Borderline Pinot Grigio (same grape) vs. Quest Riesing would be good, except they're sold out of the Quest.
That's why I suggested Alsace--less of a challenge to get a matched pair.
re: Robert Lauriston
I like the idea of comparing the two varietals with wines from the same region. Seems the best way to see their similarities/differences, as different climates and terroirs would confuse the issue if you compared, say, a Mosel-Saar-Ruwer Riesling with a California Pinot Gris.
That said- why not try one of each from Austria? A quick search on www.wine-searcher.com pulls up lots of options for each.