Can you play all piano music on a keyboard? What other considerations are there?
Most music for piano will also be possible to play on a keyboard. How different the experience will be, really depends on your keyboard. It is also a question of your preferences and needs.
Here are some important differences to consider:
Number of keys
Many keyboards will have fewer keys than a piano. See this question for good answers to what you will lose with fewer keys (61-key keyboard vs 88-key piano). For most musical pieces this will not be a problem, but there are some composers that has written pieces that use the full extent of the 88-keys (see the previous link for examples).
What you normally call "keyboard" will often not have weighted keys, which means you lose some dynamic in volume and intensity. You do have digital pianos that have weighted keys, which gives a natural feeling and dynamics control, but this is not so common in keyboards. A digital piano is designed to have a more natural sound and to feel like a real piano, but it is also more expensive. See this question for some good answers to how having weighted keys influences your playing and sound.
Keyboards come in all price categories, and of course you will see big differences in the most expensive vs. the cheapest. The more expensive, the better the sound is usually the general rule here. If you aim for the most natural piano sound and feel and are willing to spend money, you should go for a digital piano, which are more expensive than most keyboards.
Obviously, pianos are hard to move and often require specialized piano movers.
"Digital pianos" with everything built in are meant to be furniture. They're easier to move, but you still won't be taking them to your local jam session on weekends. Instruments meant to be used with separate stand, amp, and speakers are sometimes called "stage pianos". They can still be a challenge: mine is about 50 pounds, just barely luggable by one person, and slightly too wide to fit across most back seats. Models with fewer keys or unweighted actions can be very portable, but you lose the range and easy control over dynamics.
Dynamics: on both piano and keyboard: play a fragment of music as softly as you can, then again as loudly as you can (without changing the volume). Listen to the difference. Pianos usually have the wider range in both volume and tone color.
Sustain: play one note as loudly as you can, hold it, and time it. Keyboards may have a shorter and/or less natural-sounding sustain. (I base this mainly on experience with my 20-year-old keyboard; larger RAM for sample memory may have fixed this problem in newer models, I don't know.)
Resonance: play a note. Release it, then press down the sustain pedal, then play the note again. The latter will sound different on piano, but not on most keyboards. Also, instead of holding down the sustain pedal, hold down a chord on a piano (press the keys very slowly so the hammers don't hit the strings), then strike another key hard and release it, and you should hear the undamped strings ring faintly. Also, pianos behave differently if you lower the dampers halfway instead of releasing the pedal completely. (Newer digital pianos can simulate all of this, but I don't know how common or high-quality this feature is.) This may sound picky but if you like to play with a lot of pedal you'll notice the difference.
Also, a bit more obscure, but there are extended techniques that are only possible on a piano (and even then not on all pianos): reaching inside to pluck or mute strings, putting objects on the strings, etc., etc.
And obviously the reverse holds too: keyboards may include alternate sounds, effects, sequencers, ability to quickly transpose or tune to non-standard pitch (or even alternate temperaments), audio and midi input and output (though some of that can be done on pianos too). And there may be things that are easy to do on a light, unweighted keyboard (but I don't have examples).
Keyboards are generally easier to record, amplify, or run through external effects.
Good answer, but I need to strongly emphasize the following after playing on a digital piano (basically a weighted and better keyboard) for 10 years:
The pedal, the pedal, the pedal, is 100% completely different. It doesn't matter the first few years of playing, because you are busy getting other more "important" (or rather, more immediate) techniques down pat.
The problem is once you get more advanced, you find you are not able to play pieces as they are meant to be. Furthermore, with classical improvisation, the pedal difference will prevent you from further expressing yourself.
As stated before, the pedal acts as an on/off switch, even if it has a "half damper" feature. I think the one benenfit of having this fake pedal is that you learn quickly to become very "snappy" or "clean" with your pedaling technique <- I feel like this is because you must be "quick" and "sharp" in order to produce the desired sound, where normally on an acoustic piano, it would be different. I feel like this quick and clean skill may be helpful when playing on an acoustic, but I have nothing to base this off of.
My strong advice is to figure out a way to get an acoustic piano rather than playing off of a digital piano. Or have regular access to someone else's acoustic piano and get a digital piano.
Lots of people worry about sound quality and key weight; lots of people forget about the pedal. Hope this helps.
By the way, if the question is really the classic "what should I buy the beginning player in my house":
If they're taking traditional piano lessons with the goal of learning to play the classical piano repertoire really well, and if they've been at it more than a few years, then they need a piano in good working order.
If they're learning more as a way to learn about music in general and aren't interested in any particular genre, either can work.
If they want to play in a band, then they probably need some kind of portable keyboard.
Either way those goals need to be explained to their teacher (they should have one) and the teacher needs to agree with the choice.