I'm only a guitarist, but I play once a week on a real piano at my teacher's studio, and I have a Yamaha P-115 digital piano at home (weighted keys, $600 retail most places -- I got lucky and paid $400 at a salvage store). The feel is a bit different between the two, but it's in the ballpark (for a tyro like me). No trouble adjusting. I originally started with an unweighted "synth action" MIDI controller keyboard, and that was different enough to be a problem. It only has 47 keys or something too. Just not a good idea all around.
I asked my teacher for recommendations in my price range. He gave me a short list of instruments with weighted keys, with some pros and cons for each one. I would recommend that you talk to your teacher (or some other experienced person, if you don't have a teacher yet) and ask for advice. Be prepared with your best current understanding of your needs and desires, even though you'll learn more about your needs and desires after the decision is irrevocable. I've found that knowledgeable people in a lot of fields are amazingly generous with thoughtful advice for newbies. Mind you, this doesn't include the salesman at the music store or the motorcycle dealership; they have a conflict of interest.
Aside from keyboard feel, there are pros and cons to the P115: The P115 is in better tune than my teacher's beater piano, works with headphones, plays its metronome through the headphones, takes up relatively little room, can be stuffed in a closet, speaks MIDI to GarageBand, and has other sounds besides piano, which I sometimes use. Of course, tuning is easy to fix if you own the instrument, and the other points may be totally irrelevant to you. I started piano lessons to learn to play keyboards, not to be a pianist.
There's one big downside to the P115: Play a note, and hold it down until it decays to nothing. Do it again with the same note. It's the same. It's not just the same pitch, it's the same exact recording. On a physical piano, they will never be precisely the same twice (especially on notes with more than one string, which will produce audible interference patterns). But if you play the same sample twice in a row, it will be precisely the same, every... single... time. For me, there is noticeably artificial and distracting at the end of long sustained notes -- and that's my guess at what it might be. But I don't spend all that much of my practice time listening to the very ends of long sustained notes.
Smart engineers can fix that. If they haven't yet, they will. On a $1000+ instrument, they may have done it already. I can't say, I haven't played any $1000+ digital pianos. If you're in the market for one, I urge you to try letting a note die away to nothing, while listening with headphones, and then repeat a few times. See if it sounds fake to your ears. Then decide whether that's a deal-breaker anyhow. It isn't one for me.
Through headphones, the total absence of any extraneous sounds out of the instrument is also weird and creepy, and the built-in speakers are... OK, if you don't expect miracles. They sound better if you turn it up a bit. Again, a $600 instrument isn't a $1000+ instrument, and any given $1000+ instrument isn't any other $1000+ instrument.
But the only way to choose an instrument for you is for you to sit down in front of it (or pick it up, or sit in it, behind it, or under it) and play it. Weighted keys are a must. The rest is a matter of what meets your own unique needs. There is no general answer to the question of which instrument to buy. That's why there are so many different instruments out there. Is a saxophone better than a Hammond? Well, I dunno: Are you trying to sound like Benny Goodman, or Jeff Porcaro?
All things considered, I'm satisfied with my own choice, given my needs, desires, the high cost of my other hobbies, and the finite size of my living room.