To answer this question, let's first imagine what a complex organism would be like without cellular differentiation. An undifferentiated cell would be something like a stem cell, a generic cell that could differentiate into anything but hasn't yet. If you had a lot of these cells growing together, you might have something like a sphere. This is okay as long as the sphere is only a few cells. Each cell has enough surface area with the environment to take in gasses, water, and nutrients while expelling waste. If the sphere became much larger, like billions or trillions of cells, this would no longer be the case. The cells inside the sphere would be trapped behind other cells, and they would have to rely increasingly on slow diffusion. This would not be sustainable.
Okay, so a sphere wouldn't be the best shape. What if the cells spread out, like a mat? (This is actually what bacteria does when it forms a biofilm) Now the surface area problem is solved, but it can't do very much beyond some chemical exchange. What if we want our organism to swim, or run, or have an easier time distributing its offspring? Individual undifferentiated cells can crawl, but they're pretty slow. Can this be done faster?
Yes it can! For dispersing offspring, we can do something like have part of our organism high off the ground, with offspring wrapped in specialized cells that give it lots of nutrients and protect it from the elements without increasing weight so much the offspring can't be carried off by wind. To move quickly, we could have elastic and quickly lengthening muscle cells anchored by bone that allows for mechanical movement. We could even make these cells so specialized for being elastic or structural that they sacrifice metabolic function, and carry around some cells that are extra good at metabolizing with them. And to solve the surface area problem as our organism gets bigger, we could have carry around some cells spread throughout the organism that are very thin and good at transporting resources (even in the shape of tubes that can carry water).
So to sum things up, cellular differentiation leads to a variety of specialized cells that can let the organism do complicated things.
