So for example if I am flying the Boeing 737-800, the FMC gives me three flap settings and approach speeds.
Why would a pilot want to fly with anything less than full flaps where possible?
Less flaps gives a faster approach speed. In some scenarios, a faster approach speed is better than a slow one, for example:
A higher approach speed provides a better stall margin and higher control authority in challenging situations. For example, the Boeing 737 QRH calls for a flaps 15 landing in "One Engine Inoperative" and "Stabilizer Trim Inoperative" scenarios.
In the crosswind scenario, let's assume your approach speed is 100 knots, and the crosswind is 20 knots. The crosswind is therefore 1/5 of the approach speed, and you need a larger crab angle to keep the airplane aligned with the runway. If you increase your approach speed to 120 knots, then the crosswind is only 1/6 of the approach speed, and the crab angle is smaller, making landing easier.
In strong crosswind-conditions you want to consider not using full flaps to give the wind less attack area. So you have a slightly higher approach and landing speed, but you aren't blown as much to the side as you would be with full flaps.
In a Cessna 172, for example, this is normal practice.
In a Cessna 150 for example, you want to be on the downwind going 70-75 knots, 65-70 on base, and 55-65 on final, depending on wind gusts. A pilot in a small plane might tend to put in 10° of flaps on the downwind, 10° on base, and adjustments or anything extra on final to allow them to get their speed to where they need it. It is nice to be in the middle where you can add or remove some if necessary (assuming you're not too close to the ground).