Within the bounds of normal reason, and providing that the Flap settings to be used are approved and certified for Takeoff -
(1) A higher Flap setting will provide for a shorter Takeoff roll, but a decreased 1st and 2nd Segment Climb gradient. Even though acceleration will be slightly less, TODR will still be lower because the V1/Vr/V2 to be achieved are lower. An additional advantage is that because the aircraft is airborne earlier, clearance of "close in" obstacles will be improved, even at the lower climb gradients available.
(2) A lower Flap setting will provide for a longer Takeoff roll, but an increased 1st and 2nd Segment Climb gradient. Even though acceleration will be slightly higher, TODR will still be higher because the V1/Vr/V2 to be achieved are also higher. Because the aircraft is airborne later, clearance of "close in" obstacles will be degraded, but clearance of more distant obstacles improved because of the higher climb gradients available.
Somewhere along the way, in the 1st or 2nd segment, the 2 possible flight paths intersect, where both flap settings are equal.
As a general rule, if Accelerate-Stop or "close in" obstacles are the more limiting, use the higher Flap setting. If more distant obstacles are most limiting, and Accelerate-Stop or "close in" obstacles are not limiting, use the lower Flap setting.
There are cases for the sometimes quoted "infinitely long runway with no obstacles" where the higher Flap setting will actually lift more weight than the lower Flap setting. This arises, particularly in down-wind Takeoff cases, where operation with the lower Flap setting hits the Brake Energy limit first, whereas (for one aircraft that I do the number crunching for), Vmbe doesn't come into play at all at the higher Flap setting, due to the lower Takeoff speeds (V1 in particular) at all.