I have had both lenses and still have the DA*300. The DA55-300 will suffice for most as its IQ is great for a consumer type zoom lens at a great price. With the DA*300 :
1.Weather sealing
2. F4 that will allow use with some teleconverters. I use the AFA 1.7x to reach to 510mm.
3. Quiet SDM
4. Great build quality
5. Better IQ
With the DA 55-300, you get:
1. Smaller and lighter. Easier to take on travel
2. Lower cost
3. Build quality that is not as good
4. Too slow to use with most teleconverters. F5.8 at 300mm
5. Noisy focusing and focus hunts like crazy sometimes.
Until you really get into premium glass, you can consider the DA55-300 a very decent bargain since most consumer grade zooms do not behave as well as that lens. They generally have some Achilles heel that has to be dealt with. The problems with the 55-300 are rather minimal compared to Sigma and Tamron 70-300mm zooms.
The DA*300 is a top shelf piece of glass and its sharpness IQ is important to those who seek the best they can get within the Pentax system. There is also the FA 300mm f4.5 that can sometimes be had for a little less with IQ just as good as the DA*300. but is without SDM focusing and weather sealing.
Also, I would offer that the longer lenses are not always about bringing farther subjects closer but also for taking close ups of closer subjects in order for fine detail to be enjoyed. When I lived in Louisiana, most of my bird and gator shots were done within 100 ft. Sometimes much closer. It is those pictures that I find most pleasing because of the detail in the feathers, skin or whatever that is picked up.
This was taken with the 55-300mm at 230mm at about 33ft
Great Egret Reflection - Pentax User Photo Gallery
and this one with the DA*300mm at about 6 ft
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https://www.pentaxforums.com/gallery/images/20990/large/1_IMGP0248.jpg