This is my understanding:
When it is decided that you are going to stay at least one night in the hospital, you are "admitted". This could intail any or all of the following: checking in or registering, being taken to a room, or just being told "We'd like to admit you", etc.
Once you've been "admitted", it could be said that you've been "hospitalized". The morning after you've been "admitted", someone may call your work and say: "I just wanted to let you know, ____ won't be at work today because he/she has been "hospitalized"." The person on the other end of the phone may then say: "Oh my gosh, what happened?" The person calling on your behalf would then say something like "He/she had _____ pain last night. I took him/her to the emergency room, and they "admitted" him/her."
I hope this helps!
