While usually "interesuję się" means the same as "jestem zainteresowany/a", in this case a native speaker would understand "jestem zainteresowany nauką" as "I'm interested in learning", with the subject implied by the context. For example, it's something you'd say to a representative of a school/university you would like to start attending.

The reason is that "interesuję się" implies a hobby, and "jestem zainteresowany/a" implies something that piqued your interest just a moment ago.

For full comparison:

"interesuję się" – about hobbies, about things you're investigating, about long-running events and trends you're following, about a field you study
"jestem zainteresowany/a" – about a recently offered offer, about a thing you want to purchase
"interesuje mnie" – about an offer, about a field of study
"ciekawi mnie" – about a curious event or fact, about things that bug you and you need to learn about them

Some loose translations:

Interesuję się filmami Tarantino. – Tarantino movies are my hobby.
Jestem zainteresowany filmami Tarantino. – I'd like to see some Tarantino movies.
Interesują mnie filmy Tarantino. – I find Tarantino movies interesting.
Ciekawią mnie filmy Tarantino. – I'm curious about Tarantino movies.

Of course the best way to actually learn those differences is to see how those phrases are used in the wild.