The difference that matters is not how they look. It is that acrylic becomes waterproof when it dries and watercolor does not.
Almost every practical difference between the two follows from that one fact, and it is the thing most comparisons leave out entirely.
What each one is

Watercolor is pigment bound in gum arabic, a water-soluble gum. Thin it with water and it stays soluble forever. A dried watercolor painting can be re-wetted and moved around years later.
Acrylic is pigment suspended in an acrylic polymer emulsion. It thins with water while wet, then the polymer particles fuse as it dries into a flexible plastic film that water no longer touches.
Everything below comes back to that.
Transparency, and where the white comes from
Watercolor is transparent. Light passes through the paint, hits the white paper, and comes back, which is what gives watercolor its particular glow. There is no white paint doing the work.
The consequence is that you have to plan your highlights before you start. Whites are the paper you deliberately leave unpainted. Paint over a spot you needed to keep white and you cannot get it back, which is why watercolorists think several steps ahead.
Acrylic is opaque by default and you paint the highlights on last, in white, over whatever is underneath. It can be thinned to behave transparently if you want that, which makes it the more flexible of the two, though a thinned acrylic wash never quite matches the luminosity of real watercolor.
Corrections
The biggest practical difference for a beginner.
Acrylic forgives almost everything. Wrong color, wrong shape, wrong value: let it dry, paint over it. The layer beneath is sealed and inert, so it will not bleed through or lift. You can rework a painting indefinitely.
Watercolor forgives very little. You can lift some pigment while it is still damp with a clean brush or tissue, and some colors lift reasonably well after drying. But many pigments are staining and will not come out at all, and scrubbing damages the paper surface so it never takes paint evenly again.
In watercolor, a mistake usually has to be absorbed into the painting rather than removed. Some people find that liberating and some find it maddening, and it is worth knowing which you are before you buy.
Layering
Acrylic layers cleanly. Because each dried layer is waterproof, you can put anything over anything without disturbing it.
Watercolor glazing is more delicate. Adding a transparent wash over a dry one will partially re-dissolve the layer underneath if you work it, so you lay the wash down in one confident pass and leave it alone. Fiddling is what turns a glaze muddy.
Drying and working time
Both dry fast, and both punish you for it in different ways.
Acrylic is touch-dry in minutes and fully cured over days. On the canvas that means quick progress; on the palette it means a color you mixed twenty minutes ago may be unusable. Retarder mediums, a damp palette or a light mist of water all extend the window.
Watercolor dries quickly too, but because it re-wets, the paint on your palette never really goes to waste. A dried-out watercolor pan is simply a pan, and a drop of water brings it straight back. Nothing is lost.
Brushes, which is a real difference
An underrated advantage of watercolor: it will not ruin your brushes. Leave a watercolor brush to dry with paint in it and a rinse restores it, because the pigment re-dissolves.
Acrylic hardens inside the ferrule and destroys brushes permanently. Rinse between colors, wash properly at the end, and never leave a brush standing in the water jar, where the bristles bend against the bottom and the soaking loosens the adhesive.
Surfaces and setup
Watercolor needs proper watercolor paper, and this is not optional. Ordinary paper buckles and the paint sits in puddles. Heavier paper resists cockling, and stretching or taping the sheet down helps further. It also needs a near-horizontal surface or the wash runs to the bottom edge, which is the one genuine reason to own a convertible easel. Resist belongs to the watercolor side of that split and does not transfer to acrylic.
Acrylic goes on almost anything: canvas, board, paper, wood, most primed surfaces. It is normally worked upright, where you see the painting the way a viewer eventually will.
On portability, watercolor wins comfortably. A pan set, a water brush and a pad fit in a coat pocket. Acrylic means tubes, a palette, a water jar and a rigid surface.
Finishing and framing
Acrylic can be varnished, which evens out the patchy sheen different pigments dry to, and it hangs perfectly well unglazed.
Watercolor must be framed under glass, without exception. The surface stays water-soluble indefinitely, so a splash or damp air can damage it years later. That is an ongoing cost worth factoring in, since glazed framing is not cheap.
Which to choose
Choose acrylic if you want to be able to correct things, you like working opaquely and building up, you want to paint on canvas and hang it without glass, or you are starting out and want the more forgiving medium. It is the easier first choice for most people, and it is what we would suggest.
Choose watercolor if you are drawn to transparency and light, you want something genuinely portable, you enjoy working quickly and decisively, or you like a medium that keeps some control for itself. It is cheap to start and hard to master, which is a fair description of its appeal.
They are not in competition and plenty of people do both. But learning one properly beats being roughly capable at two, because most of what you learn is how that paint behaves.
What paint by number kits use
Acrylic, essentially always, and the reasons follow directly from everything above. It covers printed numbers opaquely, which watercolor could never do. It dries fast enough to work across a canvas in one sitting. It cleans up with water and needs no solvents. And when you finish, the canvas can be varnished and hung without glass.
You will occasionally see kits sold as oil painting kits. Nearly all of them contain acrylic, and if the instructions say to clean the brushes with water, it is acrylic whatever the box says.
Ours are acrylic, vacuum-sealed so they arrive moist, with every color ready to use and nothing to mix. If you want to get a feel for how acrylic behaves without buying a palette full of tubes and deciding what to paint, a kit is a reasonable way in.
Our catalog is licensed original paintings, and there is a replacement paint set for every design. For more on how the different mediums compare, see our guide to paint types, and for handling acrylic's short working window, paint by numbers blending.