If you want to get into a private building or part of a building, you seek admittance. Admittance is a formal word and is sometimes used on signs.
1. How then was he to gain admittance?
2. Bernstein opened a door marked NO ADMITTANCE.
Admission has the same meaning, but is less formal. You do not usually use ‘admission’ on signs.
3. Marsha was insisting on admission to David’s office.
4. To gain admission, one had to ring that bell at the main gate.
You always use admission when you want to talk about going into public places such as theatres and museums, or into a hospital as a patient, or being allowed into a university as a student.
5. ..free admission to all national museums and galleries. *
6. …priority cases for admission to hospital.
7. He applied for admission to Harvard.
• An admission is a confession, usually made rather reluctantly, that you have done something bad, unpleasant, or embarrassing.
8. The admission of guilt is hard.
9. They made no admission that the newspaper had been fooling the public.