I know Get Axis works fine for what you're trying to do. I haven't tried Get Axis Vector, perhaps it acts differently.
Technically, you shouldn't need it. You can Get Axis and it will figure positive and negative values, you just need to get each axis you intend to move on if that is sensible for your situation.
Like above, Gravity in the axis input settings determines how fast it falls back to zero (no input) after the player releases the input key. If Gravity is set to zero then the number will not reduce unless the negative input key is pressed. Otherwise, Gravity is the rate at which the value falls back to zero when the input key is released. Get Axis grabs the result value, so you can use it to manipulate or multiply forces that influence the player position this way, like using Add Force, for instance, if based on the Axis Input could be a multiplier to add a certain amount of force every frame and the Input setting is simple the throttle, while sensitivity is the rate the throttle can increase, and gravity the control for it falling back to neutral.
Additionally if Snap is checked on then when the negative key is pressed the input value will snap to zero and begin going the opposite value direction, if you turn snap off it will fall at the sensitivity rate toward the negative input value.
Hope that makes sense...
You might check out the Oomph project in the community examples thread for a more elaborate example of how Get Axis can work on a large scale and how tuning the input settings makes a big difference.