In my experience, the water-born varnishes generally were less durable, required more coats, and did not bond to previous coats as well. Most of these are currently referred to as a polycrylic finish and some are available as 'wiping finish'. Polyurethane is a different animal involving oils and polymers, I'm not a chemist and am not about to try to explain, except the oil based finish bonds better, builds better, and is harder than the water based finish. It also smells worse, is flammable, and requires mineral spirits to thin or clean up.
The polyurethane comes as a brush/roll on, wipe on, or even a rattle can spray. I prefer to use wipe on oil finishes for small projects. Wipe ons are easier for a less experienced user to get a good finish. Since it has been several years since I used a water based varnish, I would recommend doing your own research and testing if you want to go that way.
The wiping polyurethane is basically applied like any penetrating oil finish. Read and follow the instructions. There are web and YouTubes for further learning.
There are several wiping Polyurethanes on the market, Minwax and Watco are the most common and are good finishes. I have Minwax in a gloss finish and in a satin finish in my shop right now.
Generally, you scuff sand between every 1 or 2 coats on any of these finishes. You'll lay down 4 or more coats to acheive good results.
Practicing on sample pieces is always a good idea.
Ace, Lowes, Menards and Home Depot are good common sources.
Hope I've added a little clarity to my earlier response.