Technically polyurethane is a kind of varnish. Varnishes are processed mixtures of oil and resin. Polyurethanes use synthetic resins, while what is usually called "varnish" uses natural resin. That aside, I'm not clear on what your problem is. Is it that you've changed your mind and now want a higher gloss? Did you expect a thicker, smoother coating? Three coats is usually plenty.

I guess I expected a thicker smoother coating. It still looks like raw wood to me...and still feels more like raw wood. I'm not sure what the difference was but in the store when we tried it on a sample of cabinet wood (but on what should have been an unfinished side) it felt smoother and felt "coated" so I was put off by what it feels like now. I just wanted to make sure we hadn't done something wrong I guess. It would be a shame if it got ruined because we didn't finish it properly...

For future reference, is there any advantage to one or the other?

Three coats is usually sufficient for protection, but won't give you much "build". If you want a thicker, smoother coating, try a rubbing polyurethane like Minwax. Poly is thicker than varnish, so each coat of poly is roughly equivalent to 2-3 coats of varnish, in my experience. You should be able to apply the poly directly over the varnish after a light sanding.

First question is---Did you use a 'clear' varnish? Oil based finishes generally add an amber tone to whatever is finished.

Water based and 'clear' oil based finishes are rather bland looking.

Wiping finishes by design are thinner than finishes intended to be applied with a brush or roller----necessitating twice the number of coats to achieve the same general protection/color of regular finishes.

In your case, I would get a small can of oil based satin varnish and add one coat to the legs to see if that fixes the color/quality of the finish.

The explanation Jon gave is correct---and three coats of any regular(non-wiping) varnish is the normal recommendation.

For what you are doing both finishes will work fine, most spray applied finishes are better but oh well. If your coats are rough it is because you aren't sanding between coats, 280 - 400 grit. Or you're getting airborne dust which is common with slow dry finishes such as you are using.

Regarding varnish vs polyurethane, I don't think you necessarily chose the wrong product. I'm curious about the dull finish you say you selected. I'm also curious to know if you sanded between coats. Varnish takes a while to dry, so it tends to collect bits of airborne dust, making the surface feel slightly rough when it's dry. Also, if it is a dull finish, I suspect it requires thorough and frequent stirring to keep the flattening agents suspended. This can introduce airbubbles that will also contribute to a less than smooth finish. By the description, dull finish would seem to indicate to me that it won't produce a shiny gloss finish, regardless of how many coats you apply. Would you consider a semi-gloss or gloss finish?

I would first do a light sanding of the surface with a fine grit sandpaper. Nothing coarser than 220, higher grit if you have it available. Once the surface feels smooth, make sure it's wiped very clean of all dust and apply a coat of either semi-gloss or gloss varnish. Hopefully this will give you the look and feel you are trying to achieve. Good luck.

Sanding sealers are typically used with varnish due to the thinness of the coating. This eliminates raising of the grain as new layers of varnish melt a bit of the previous coat raising the grain even after sanding between coats.
Poly does not do this amd does not require sanding sealer but does require scuffing between coats as it does not melt a bit of the previous coat.