As you explained in the start of your question, osmosis helps cells maintain an equilibrium with regard to the amount of water inside and outside of the cell. When a concentration gradient of a solute (such as salt) exists on either side of the membrane, water will diffuse to the side with a higher concentration of solute until the concentrations on either side of the membrane are equal. Once the concentrations are equal, water flows back and forth through the membrane at the same rate and the concentrations are maintained.
Osmosis is vital to the growth and development of cells because it allows the cell to adjust to changes in the environment. Under normal conditions, osmosis allows cells to maintain an equilibrium inside and outside the cell membrane. However, extreme conditions can sometimes cause cell death as a result of osmosis.
For example, when cells are exposed to a solution with a high concentration of solute (these solutions are called hypertonic) water rushes out of the cell towards the higher concentration outside and the cell can dehydrate. This is why we aren't able to drink seawater. The salt concentration in the water actually dehydrates our cells rather than hydrating them.
Conversely, cells exposed to hypotonic solutions (solutions that are more dilute than the internal environment of the cell) take on water so rapidly that they swell and burst.
Different organisms have evolved different ways of dealing with very hypertonic or hypotonic solutions. In animals, the blood works to maintain an isotonic (when the solute concentration is equal inside and outside of cells) environment. In plants, the cell wall limits the amount of water taken in by the cell and prevents bursting.
In plants, osmosis not only helps maintain proper water levels in the cell, but it also allows plants to take up water from the soil into their roots. Through active transport, plants keep a high concentration of solutes in their roots. This gradient causes water to be drawn into the roots, and it can then be transported through the rest of the plant.