In case of a moving bus, we are also moving with it ,ie, we are in motion. And due to inertia of our body, our body try to remain in motion even when the bus brakes to stop resulting in falling our body forward.
And in case of a bus at rest , we are also at rest (as we are in the bus). And due to inertia of our body, our body try to remain in rest, that's why we fall backwards when the bus accelerates from rest.
I hope it will help you.
- Mayan Anand