I have the opinion that Royal Enfield and Harley Davidson occupy much the same position in their relative markets, both building machines based heavily on tradition, and both having a die hard following in their home countries. One will hear the same mentions of "soul" and "feeling" in discussions about what is great about the makes.
I'll avoid that area and concentrate on discussion of the mechanicals. Both are air cooled, pushrod engines. You didn't say, but I'm assuming that you are riding the 883 Iron. In the US, we also have a 1200 cc version. In 2016 the Iron 883 would have had a 5 speed transmission and belt drive.
Let's discuss what would change if you switched to an RE Classic 500.
You'll be giving up considerable torque and horsepower, stepping down from about 49 hp and 54 ft-lbs to around 27hp and 31ft-lbs. I consider that amount of power quite adequate for what I use my 2012 C5 Desert Storm for, but it does somewhat limit the machine for highway use. My C5 maxes out at about 120 kph. Your HD can probably go 150 kph or more.
( I regularly ride with a friend who rides an Ironhead HD Sportster, and I'm always out in front, waiting for him, but this has much more to do with riding style and confidence, and not much to do with the machine's capabilities at all. )
In exchange for the lower power, you will get substantially better fuel economy and range. Both machines have gas tanks about the same size. Since the RE can get around 30kpl, to the HD's 20kpl, you will be able to ride considerably further on the same amount of fuel.
The RE, is shorter, it's wheelbase is only 1360mm vs the HD 883 at 1515mm, but it is taller, and the seat is several inches taller. For this reason, if you are a shorter person, you may find the RE difficult to manage at first, even though it is easily 55kg lighter than the HD.
Another difference that you may notice is in the handling. I think that the HD's considerable more front fork rake, 30 degrees vs the 26 degrees of the RE C5, is largely responsible for this difference, along with the difference in wheelbase.
I have owned and worked on Harleys, and own the 2012 C5 I mention, and consider them to have similar quality of materials and engineering. I consider the RE C5 to have excellent engineering in terms of it's ease of maintenance and repair-ability, better than any Harley I've ever worked on. In my opinion, both makes are much better in this respect, than the Japanese bikes I've worked on, which tend to be engineering for more sophistication, rather than simplicity and robustness.
If I were you, even if I bought another machine, I would consider hanging on to your Harley for a bit longer. You may have heard, Harley Davidson announced some months ago that they were closing their factory in India, where the Iron 883s for the Euro market were being assembled. I expect that this will result in the value of HD bikes in Europe to increase, but this may take a few years while bikes in available stock are depleted. So that Harley Iron, if you can keep it in good condition, may appreciate considerably in value, vs what it is worth today.