This is easy enough to test. Take a tire and fill it. Test the pressure. Let it sit outside overnight on a cold night and test it again. The pressure will be lower.

Consider this, as well: yes, it’s about differential pressure, but it’s also volumes. “Pressure” is the outward force from all of the molecules in the air bouncing around in a space. As temperature increases, those molecules move faster and bounce more. Since pressure is just us measuring that bouncing, increased temperature increases pressure.

Why doesn’t the atmosphere increasing in temp balance it out? Size. The tire is SIGNIFICANTLY smaller. This means each individual molecule only has to go a little bit before it’s hitting another wall. Compared to outside the tire, those same molecules bounce off everything, some ricocheting into the tire, and others in whatever random other direction. The change in movement speed of the particles may be the same, but the change in how frequently they’re smashing into the tire is different, thus different pressures.