In New York State, the New York State High School Equivalency Diploma (HSE) is functionally the same as a GED. They both fulfill the same purpose: demonstrating high school-level competency for those who didn't graduate high school traditionally. The difference is simply in the name and the testing provider. The New York State Education Department uses its own test (the NYSHSE test) while other states might use the GED test.
For college admission, there's no difference between an HSE and a GED. Colleges will accept either as proof of high school equivalency. The important factor is having the credential itself, not which specific test you took to earn it. Your academic record and other application materials will carry more weight in the college admissions process than the specific name of your high school equivalency diploma.