I'm a bit surprised by the advice to use RR Ties. But in could just be what's available in different areas of the country. Here, they are made out of just about any type of wood, but mostly junk trees that are not worth milling into quality lumber. Gum trees are probably used most o fall. RR Ties are not meant to be put in the ground, they are placed on top of well drained gravel. In my experience, they rot out very quickly. With posts, the entire wood in the ground gets eaten up by termites and other bugs once they find a way through the tar that coats it. Remember, it's not pressure treated, just coated. In landscaping, it disappears from the ground side and for years, it looks fine, but there isn't anything behind what you see.
As for the original question, round or square, the biggest difference is cost. Round posts cost a lot less money. Be sure to buy them when there are a lot to choose from. If you show up and there are only a few left, you know all the best ones have already been sold and you are left with the rejects. In my area, you can buy round posts at a dozen different places, so there is always somebody with some good ones, somewhere.
Treated posts do not rot in the ground or in the air. They always rot at ground level. This is where the moisture remains, and it's where the grass shades it so it remains just a little longer.
The trick to protecting a post in the ground is to have the water flowing away from it. Concrete has an advantage because you can build it up the sides of the post with it and never have to worry about it again. With soil compacted around the post, no matter how much effort you put into compacting it, it will always be looser then the virgin soil around it. Over the years, it will absorb more moisture in that soil, but that's not the real issue. That soil will also settle and get washed away around the post rather quickly, and you are left with a low spot that holds water. This is why wood posts rot!!!!!!
For a gate, corner posts and all bracing, I use concrete. It's the only thing that after you are done, you are actually done and you don't have to mess with it again.