What causes 'route to' to be faster is that rivers connect cities! I was completely perplexed by a 'connected to trade route icon' (the one that's like the recycle sign) on a brand new city I dropped with no roads nearby.
This feature doesnt seem to be documented anywhere, but is definitely true. If you connect a city, resource tile, etc to a river, and that river connects to a city, the two will be connected for trade purposes. It's really convenient if you know about it - you can expand along a river, for instance, and your new cities will have the same luxury and strategic resources.
As far as I can tell, religion doesn't spread this way.
Also, that's what causes the strange appearance at rivers before bridges - roads turn into little docks.
However, I totally agree about the appearance of roads in general. The first road tile I ever built turned into a ridiculous S-curve and I was like

- I really dislike that roads don't take straighter, simpler paths.