For starters, interior design usually sits inside a design degree. I've taught both, and the truth is that interior design education is nowhere near as hard as architecture, for good reason.
Two different educations
Interior design is taught under a general design degree, and design as a whole is broad. So you get a broad, fun degree, but one that isn't quite serious enough. Its tutors come from a three-to-four-year degree, while architecture tutors studied for five or six years before they taught.
Architecture is too general as well, but in a way that misses the point. Even so, it's still taught in a deeper way than interior design.
Something and nothing
The deeper difference is what you design with. Design teaches you to design with "something": objects, finishes, furniture. Architecture teaches you to design with "something" and "nothing": the spaces themselves and the things within them, along with time, culture, environment and how people behave. Architects look at the largest picture as well as the smallest scales.
Whichever you choose
Whichever you choose, neither degree will teach you the design process properly. You'll need to learn how to think about design yourself, and that's what we teach.
Next step: Talk it through in a free introduction, or see Student Coaching.