I think it depends what you are looking for in your figures. While Palisades was great for the time, many of the figures are fairly outdated in terms of articulation. The paint apps are far superior in most cases and the sculpts are mostly spot on, but there are some misses and often with the bigger name characters like Piggy and Fozzie. You also have your home runs like the series 5 Gonzo. Accessories abound, but the scale is its own animal that doesn't really work with other lines (if you care about that kind of thing). They also take up a lot of space because of how large they are.
DST has far better articulation with minimal sculpt compromise. The likenesses are better for characters Palisades didn't quite get right like Piggy, Fozzie, Scooter, Rowlf, Crazy Harry, Beaker, etc. The scale works with 1:10th scale figures, which is why DST and NECA tend to make so you can get a variety of human figures in scale with the Muppets. Some likenesses are worse though, like Chef, Kermit, Gonzo. Characters with closed jackets have them closed which is accurate. Palisades left them open in many cases to show a more interesting, though less accurate outfit. DST also gave us stuff Palisades didn't. A balcony, an upright piano, Bean, FooFoo. There is a lot of good with the DST stuff.
Personally, I like to combine Palisades and DST. The DST figures are perfect for the Palisades playsets. The range of motion of the DST figures is great and the size is better for someone who lacks space. My backstage set is full of DST figures, without them being too crowded.