Trailing edge devices such as flaps increase lift by raising lift coefficient at s given AoA. Leading edge devices such as slats increase lift by raising stall AoA.
The former is nice because it allows the nose to point lower and give the pilot a better view during landings and takeoffs, and increase tailstrike clearance. The latter does the opposite.
Why have both on aircraft, as is done today? Why not just use bigger flaps? Wouldn't that be safer and simpler?
Slats increase the stall angle, flaps do not, that is the key issue - safety margin. Plus on aeroplanes with high wing loading even triple slotted fowler flaps may not cut the mustard, slats are simply required additionally.
Pic from this answer, showing that the stall AoA actually decreases in this example. The second pic in the referenced answer shows the curve with both slats extended.
There are aeroplanes with only flaps, the Fokker 28 was one.
Slats are a necessity when the wing uses slotted flaps or Fowler flaps for maximum lift. Those flaps cannot reach over the whole span of the wing to leave room near the wingtips for ailerons.
Once the flaps are extended, they increase the incidence of the wing along the flap span. This leaves the outer wing at a lower incidence, but in the vortex field of the inner wing. This vortex field induces a high local angle of attack over the whole wingspan, which would produce a local stall at the wingtip, where the ailerons prevent the use of slotted flaps. This stall would in most cases be asymmetric: One wing tip loses lift while the other barely manages to get along with the high angle of attack. The result is a strong rolling moment which will even increase once the aircraft starts to roll, because the local angle of attack at the down-moving wing will increase further. At this condition, the ailerons are of little help, so the roll cannot be controlled.
Now consider when flaps are used most often: Close to the ground, during take-off and landing. Then imagine what happens when the airplane rolls over one wing uncontrollably. To avoid this, slats are very helpful: They give the wing an extra stall margin by shifting the stall angle up.
Aircraft with lower wing loading normally use less powerful flaps, like simple camber flaps or split flaps. Since they change circulation less, stall on the outer wing can be enough delayed by washout and reduced taper, so leading edge devices can be avoided.
Because the angle of attack sensor is mounted on the fuselage, and the pilot is usually there too. Changing both front and back allows the chord AoA to remain the same.
Flaps are valuable on approach because they provide greater forward visibility for the same AoA, literally "drooping" the nose. The pilot must realize the plane will stall at a lower AoA with flaps deployed, especially at the higher flap settings. This is why on a go-around one must be careful not to "pull" excessively, even after full power is added.
Flaps also increase drag. The Slat greatly magnifies the forward, top suction peak, greatly improving the lift to drag ratio. Together with flaps, they create undercamber as well, also increasing lift, as seen with the DAE-21 airfoil. This form sports a L/D of over 150.
But Reynolds number is critical. On smaller models of lower Re values, flaps only may produce better results.
