I bought 12x R500 for second site over R300 for higher client density and having a second ethernet port so it could function as a passthrough.
There might be some "modest" range improvement due to it having double the beamflex antenna patterns (so slightly more accurate directionality). But I don't believe it is sold/intended as giving "longer range" over R300. Willing to be corrected by someone with both units up and running in their environment.
According to the datasheets the two have the same number of antenna patterns (64 per radio), RX Sensitivity is also similar at -100dBm. The main differences in the two AP's being client capacity and polarization diversity in the R500 over the R300. Assuming that you are running at the same power levels on either i wouldn't imagine that the R500 giving any straight line additional coverage over the R300. At 5GHz you may be able to increase coverage by changing the channel to one which allows a higher EIRP under regulation but I would urge you to consider potential implications before doing this.
Can I ask what the use case is and details around the AP installation position & configuration? You may find there are alternative ways to optimise what you have already.
The R300 is being used in a residence, and the main issue is streaming video (to iPads in particular). The video is sometimes choppy/cutting out in certain locations. The problem locations are ones where there are multiple walls, or a wall with some electronic equipment, in between the iPad and R300. The solution is probably to move the R300 (to a more centrally located ceiling), but that would involve running a PoE cable. So I was looking for an easier solution and was curious if a R500 would do better (a number of these have been showing up on eBay recently). Doesn't sound like it though. So I'll probably just bite the bullet and run the cable.