Gravity within the ship will only have to compensate for it's own acceleration. External gravity (such as from gravity assists) will apply equally to the ship and everything in it, and will have very little discernable effect. The ISS actually experiences almost full earth gravity, but because it's in freefall, it seems like zero gravity on the inside.
Acceleration will apply across the entire ship uniformly, and can be countered by tuning the angle and intensity of your gravity bubble.
Propulsion - If this isn't reactionless, you can apply a small gravitational field to the air, ground, water around you in a large area and accelerate pretty quickly without needing traction. This could potentially be a lot more efficient than existing air and watercraft propulsion, especially rockets. With a very large bubble, you could even get some thrust in space.
Hovering - Again, if it's not reactionless one can use a small, intense gravity bubble below something as a kind of frictionless multidirectional wheel.
Inertial damping - It was the original point, but it's important. highly manoeuvrable craft like fighters, drones and missiles get much faster and more manoeuvrable if they don't have to worry about ripping themselves apart.
High speed aircraft - normally these are uncomfortable due to constant turbulence, and high G forces when manoeuvring. If an aircraft could accelerate a bubble of air around it, this is not a problem as the turbulence mostly happens near the boundary of the bubble.
Pumping - A weak gravity bubble across a spaceship corridor can circulate air without inconvenient fan blades.
Weapons (launcher) - A series of high-intensity gravity bubbles in a line is essentially a railgun, with the additional bonus that what you launch will experience no g-forces, no electric current running across is, no intense magnetic field and does not have to be conductive.
Transport - Since we can apparently survive being fired out of a cannon, if we are fired into the barrel of an equally powerful cannon elsewhere we can travel by cannon. Probably a bit silly, since this enables many other wonderful vehicles.
Heavy industry - This could make lifting heavy things much easier.