Is the refraction different entering medium that has a higher index of refraction compared to entering amedium that has a lower index of refraction?

Yes

Explanation:

The speed of light when it travels through glass, diamond, etc, the light travels at different speed from the speed of light. Speed of the light in material is related to the index of refraction.

The change in speed which occurs when the light passes from one medium to the another is responsible for bending of the light which is called as refraction.

When the light goes into a medium with the higher index of the refraction, light bends towards normal. Conversely, if the light traveling goes from higher refractive index to lower refractive index, it will bend away from the normal.

Hence, the refraction is different in both the scenario.