“Admission” is a much more common word than “admittance” and is a good choice for almost all contexts. You may gain admission or admittance to a college, but you’ll probably be dealing with its admissions office.

So admittance is often seen when talking about formal settings, whereas admission is really more casual. For a fancy gala, you would use the term admittance when referring to getting into the event, but a sports event would usually just use the term admission.