The name causes most of the confusion. Water mixable oil paint is not a water-based paint, and it is not a hybrid of oil and acrylic. It is oil paint. The pigment is the same pigment, the binder is a drying oil, and the finished film is an oil film that behaves like one for the next hundred years.
What has changed is one thing only: the oil has been modified so that water will mix into it.
How it works

Ordinary linseed or safflower oil and water do not mix. Manufacturers get around this either by chemically modifying the oil itself, or by including an emulsifying agent in it, so that water can be worked into the paint rather than sitting on top of it.
Everything downstream of that follows. You can thin it with water, and you can clean up with soap and water. That is the entire proposition, and it is a bigger one than it sounds.
What actually changes
No solvents. This is the real reason to use them. Traditional oil painting means turpentine or mineral spirits in the room, on your hands and in the air. Water mixable oils remove that entirely, which matters if you paint in a spare bedroom, share the house, have a respiratory condition, or simply dislike the smell.
Cleanup. Soap and warm water, at the sink, in a couple of minutes.
Portability. No solvent means nothing that has to be transported, ventilated or disposed of carefully. They travel far better.
What does not change
This is where people are caught out.
Drying time. Oil paint dries by oxidation, not by evaporation, and modifying the binder does not alter that. Expect days for a thin film and considerably longer for a thick one. Some makers claim theirs are slightly faster than traditional oils; either way you are working in days, not the minutes acrylic gives you. If you came from acrylic, that long open working time is the point rather than a nuisance, and it is why blending in oil is so much easier.
Fat over lean. The rule still applies. Each successive layer needs to be more flexible, meaning more oil, than the one beneath, or the upper layers crack as they age. Water is the leanest thing you can add.
Varnishing. A final varnish still has to wait until the paint has cured through, which is conventionally six to twelve months.
Pigment toxicity. Cadmiums and cobalts are exactly as hazardous as they are in any other paint. Removing the solvent does not make the pigments safe to handle carelessly.
The safety point that stays, and it matters
Drying oils oxidize, and oxidation gives off heat. A pile of rags soaked in linseed or safflower oil can generate heat faster than it escapes and ignite without any flame being present. This is a documented cause of studio and workshop fires and it is not a hypothetical.
That risk is unchanged with water mixable oils, because the oil is still oil. People assume that because water cleans them up, the rags are inert. They are not.
Spread used rags out flat and singly to dry, outdoors or somewhere with air moving, before disposing of them. Or keep them in a sealed metal container with water in it. Do not ball them up in a bin.
The trade-offs
Adding too much water breaks it down. Thinning with water past a certain point disturbs the emulsion and gives a dull, weak, under-bound film. Use water for the first thin layers only, and switch to the manufacturer's own water mixable mediums for anything beyond that.
The handling feels slightly different. Many painters describe them as a little stiffer or tackier on the brush than traditional oils. It is a matter of adjustment rather than a defect, and it varies by brand.
Mixing with traditional oils is limited. They are compatible, and you can add conventional oil paint or a conventional medium, but past roughly a third by volume you lose the water mixability and are back to needing solvents. If you mix them, do it knowing that.
Brushes. Hog bristle is the usual choice, exactly as with traditional oils. Do not leave brushes standing in water: the bristles swell and splay and the handle lifts at the ferrule.
What to buy
The same advice as for any paint. A small number of good tubes beats a large set of poor ones. Titanium white plus a warm and a cool version of each primary and an earth color will mix almost anything, and two versions of each primary is the part that matters. Our piece on choosing paint sets goes through pigment load, lightfastness and why the sixty-color box is usually a trap. All of it applies here.
Any support needs priming. Acrylic gesso under oil is fine and standard. Oil under acrylic is not, and never will be.
Which to choose
If you want long open time and the way oil paint blends, and you would rather not have solvents in the house, water mixable oils are a genuinely good answer and there is no serious catch.
If you want to finish a painting in an evening, you want acrylic, because oil of any kind will still be wet tomorrow.
Worth knowing about kits, since it causes real confusion: many paint by numbers sets are advertised as oil painting by numbers and are almost always acrylic. Ours are acrylic on pre-primed cotton canvas, and there is no oil version of them. If a listing anywhere claims oil paints in a kit at that price, check before you buy.
Our own range of designs is here, and Custom Paint by Numbers works from a photograph of your own.