Why a bigger drop of water splits into many tiny droplets while falling from a height?
Why the cohesive force becomes weak when water is allowed to fall freely?
As Wikipedia says, the Cohesive forces of attraction between the liquid molecules (Weak hydrogen bonding is also possible) is the cause for Surface tension. Each molecule is pulled by the neighboring molecules resulting the net force to be zero. But, the molecules on the surface have no forces acting on them (Hence, no pulling). This causes the surface molecules to contract to a little extent.
But, the liquid drop's shape varies in presence of air or under free-fall due to gravity in air. See the possible Shapes of the liquid drop. Probably, it has a shape of a semi-sphere with a flat or curved surface at the bottom which is due to the air-flow. This also explains the difference that smaller rain drops have more surface tension and mostly they have a perfectly spherical shape, whereas comparatively larger drops have those flat or curved bottoms due to lower surface tension (since they have a larger area on the surface). Also, a raindrop has a critical limit. As it gets larger by combining with other drops, it has more possibilities for breaking
You can search for " Breaking of droplets" for more info
There's no weakening of forces. Shear stresses from the air rushing by the surface of the drop are acting on the surface, and lead to the break-up of droplets if they are sufficiently strong. There's a lot of work that has been done on this problem that's described in an extensive amount of literature. You can google "droplet breakup" if you want more details.