From the printer...
##From the printer... NonNon-post-processed surface smoothness of a print is mainly a function of the Layer height during print: it is smoother if one reduces the height of the steps taken between the layers. A very good FDM machine can achieve down to 0.05 mm layer height, but 0.1 mm is what most hobbyist machines can achieve if well dialed in. SLA printers cure resin in layers down to 25 micron reliably and some can get down to 12.5 micron - a quarter of the best achievable FDM.
Simple Postprocessing...
##Simple Postprocessing... ManyMany prints can easily be post-processed.
Sanding
###Sanding PLAPLA and ABS can be easily sanded down to somewhat smooth, but you'll need to spend time and muscle grease or go electric. A sander with 300+ grit can help a lot. If you have a rotational body, you could sand the vase on a lathe.
Putty
###Putty AA plastic putty could be smeared over the stepped surface to smooth it and then sanded smooth. I personally like Valejo plastic putty, as it doesn't shrink, but Tamya putty also is a good alternative, bonding strong.
Primer-Filler
###Primer-Filler IndependentIndependent of the base material, using a spray coating can easily fill in the notches and crannies left behind during printing. Printing in 0.15 mm and giving it a short spray coat of primer-filler twice, sanding over rough spots between did a good job of smoothing prints for me.
Vapor Smoothing
###Vapor Smoothing IfIf you chose ABS, you could use Acetone vapor to smooth over the surface of a print in almost an instant. This can achieve Super smooth results with tall layer heights, as you practically melt the plastic over, but it comes at the loss of small surface details.
Similarly, ethylacetate or propylene carbonate could be used for PLA prints, though timing might differ from the well established "a couple of seconds" for acetone vapor. Note that these two chemicals are much more expensive. ###Costs Checking
Costs
Checking prices, I was quoted about 90 €/l for (chemical grade) Ethyl Acetate and 130 €/l for (chemical grade) Propylene Carbonate via a laboratory supply. Useable Acetone (not chemical grade) comes about 3.70 €/l in home depot. For comparison: chemical-grade Acetone comes with a price tag of about 50 €/l.