Depth of cut is usually the primary factor associated with choosing a saw diameter, but certain types of saws are affected in other ways.
Besides depth of cut, one factor that impacts all types of saws is that larger blades require more materials in general to manufacture, but they also require more teeth for the same type or quality of cut. As a result, they also cost more (especially with carbide-tipped blades) because of more sharpened cutting edges. For example, 7-1/4" fine crosscut blade might have 60T and cost $20 while a comparable 10" crosscut blade may have 80T and cost more than $70.
For some types of saws, like a miter saw, the diameter of the blade also has other implications for the saw's capacity. For example, a 12" (non-sliding) miter saw can crosscut and miter a wider board than a similar 10" miter saw if the board is laid flat across the bed. (You can use the miter saw's vertical travel to squeak out a little more crosscut capacity by laying the board on its edge across the bed, with the face of the board against the miter saw's back fence.) Typically the same relationship exists for sliding miter saws, but one could certainly design a 7-1/4" sliding miter saw that has greater crosscut capacity than any given 12" sliding miter saw.
For table saws, the only implication is generally depth of cut, though if you want a table saw with a blade braking system (SawStop or Bosch's upcoming ReaXX, you're basically stuck looking at 10" saws.
Another issue you may face depending on the size you choose is that some blade sizes may not be as readily available at local stores--for instance, 9" blades for older table saws.
There are also indirect implications for other types of saws. For example, a 7-1/4" circular saw will have a smaller, lighter motor and will thus cause less fatigue on the operator than an 8-1/4" circular saw when lifting or pushing the saw over many cuts.
As long as you're using a sharp blade of the right type and with an appropriate depth of cut, the blade diameter is often largely inconsequential and it becomes more important to have the right type or size of motor in your machine. If your saw's motor does seem to get bogged down, you should confirm the blade is actually sharp and/or switch to a thin-kerf blade.