Why is it important that a cell membrane does not allow all dissolved substances to diffuse freely through it?

The overarching imperative the cell needs to follow is, basically, to maintain its identity and distinction from its environment. Life is inherently a process that depends upon highly organized and localized events that rarely or never occur in large, regulated amounts outside of those conditions. The cell needs to sustain its unique environment in order to fuel the reactions that keep it alive.

If the cell allowed everything to diffuse through the membrane freely, it would probably die promptly. A significant part of the cell's internal conditions depend upon creating concentration gradients that drive reactions; if the cell was fully permeable at all times, the effort it put into creating these gradients would be irrelevant. For example, the ATP synthase system is driven by a proton gradient; if the cell could not create the proton gradient, its ability to generate ATP would be severely diminished and unpredictable.

Additionally, the cell would probably have its electrochemical conductivity disrupted as well; on the scale of a eukaryotic organism, this would almost certainly result in near-instant death because the nervous system would become incapable of generating action potentials. This would essentially mute the system and shut down any control the brain has over the body. 

Finally, there is the simple consideration of toxicity and infection; the cell wants to avoid these things, and since they tend to be fairly large, it makes sense that the cell would be impermeable to things that meet a certain size criteria because they can safely be assumed to be a potential threat.