따다 to pick by pulling
꺾다 to pick by breaking
꽃을 따다 I imagine a scene of picking a flower by gently grabbing the whole flower blossom with my palm and pulling it or pinching the stem below the flower and pulling it.
꽃을 꺾다 I imagine a scene of picking a flower by breaking/snapping the stem.
꺾 has a meaning of bending or being bent.
E.g.
꺾쇠표 is another name of 대괄호 (braket [ ]).
cf.
Standard Korean Language Dictionary of the National Institute of Korean Language
따다:
붙어 있는 것을 잡아떼다.
to pull off something attached/sticking/adhering
꺾다:
길고 탄력이 있거나 단단한 물체를 구부려 다시 펴지지 않게 하거나 아주 끊어지게 하다.
To bend something long & elastic or hard in order to prevent it from unfolding, or to totally cut it