at vs in (the hospital) – What is different?

I like Steve’s answer; it’s probably the first distinction that comes to mind for most native speakers.

However, you’ve given us two very vague sentences. We can provide a little additional context and analyze the question even further.

Let’s say my mom is a nurse, and you had a chance meeting with her in the hallway of the hospital where she works. (Let’s assume you were there to drop off a payment in their business office.) In that case, you could use either one:

• I saw your mom in the hospital.
• I saw your mom at the hospital.

Let’s say my mother is a florist, and she supplies flowers for the hospital gift shop. She was walking back to her delivery van, and you saw her in the parking lot as you were walking in. In that case, you’d probably say:

• I saw your mom at the hospital.

because you were not in the building when you saw here, merely on the hospital grounds.

Let’s say my mom is an EMT who frequently goes on ambulance calls. You saw her getting out of the ambulance at the emergency room entrance. Most likely, you would tell me:

• I saw your mom at the hospital.

because you were outside the building when you saw her. But let’s say you didn’t see her getting out of the ambulance; instead, you saw her rolling a gurney down the hallway. In that case, you could use either one:

• I saw your mom in the hospital.
• I saw your mom at the hospital.

You might use in because you were inside the hospital building when you saw her; you might also you at because you were also on the hospital grounds when you saw her, and in sometimes conveys the meaning of being a patient, so you might avoid that, and opt to use at instead, perhaps like this:

• I saw your mom at the hospital. She was in the hallway, rolling someone on a gurney.