In one word, RECOIL.
A 28 gauge will do almost everything a 20 gauge will and won't punish your shoulder.
Generally, a scaled 28 gauge will be lighter and is a joy to carry.
I don't think either of my 20's (Ithaca Classic Doubles, William Sumners)--both at a bit over 5 3/4#--kick any harder with my 3/4 oz reloads than would a 28ga of the same weight with the same load. And you have to be careful with the "weight" thing, because as noted in above posts, many 28's are built on 20ga frames--which makes them heavier than the 20. My most recent example was when I had Browning send me a 28ga Cynergy Feather, which weighed right at 5 3/4 with 28" barrels--and which I shot quite well. I picked the 28ga rather than the 20ga, because the latter weighs about 1/4 lb less--and I thought it might be TOO light.
All of that being said, according to statistics from LODGH (the Loyal Order of Dedicated Grouse Hunters), the 28ga has really caught on with those who chase woods birds in the last 3 decades or so. Back in 1983, only 1% of those who submitted data to LODGH used the 28ga. 30 years later, that number increased to 13%. Thinking on why that happened, it occurred to me that the 28's available back in the early 80's would have been either skeet guns, or else the occasional vintage Parker or Ithaca (both VERY expensive guns!) Since then, the Spanish have sent a lot of 28ga sxs our way, and several thousand Parker Reproductions were made in that gauge. Not to mention other 28ga doubles with their barrels 90 degrees out of line (OU's).
Most grouse and woodcock hunters are interested in light guns, and they now have far more choices in the way of scaled frame 28's than they did 30 years ago.
As noted earlier, there are certainly plenty of light 20's, some well under 6#. But if you like a 28, there are also plenty of those well under 6#. So . . . why not? And the shells do have that cute thing going for them.