In the engine-out case, what is the difference between Vx and V2? Don't both achieve the best possible gradient for the configuration?
There's a slight difference.
Twin Otter figures (at max weight): V2: 80kias. Vx flap 10 (T/O config) is 75kias. So you would climb a tad quicker, but on actual engine failure on takeoff we use V2 and stick the ASI needle at 80. And since the manoeuvre allowances are +/-5kias, the POI doesn't like it when we go below 80kias on engine failure trainings
Another thing to point out of interest perhaps, it's that our SOP states that if we are climbing at Vx flap 10 (75kias) we should accelerate to V2+5 (85kias) before flap retraction. The SOP also states that our climb speed on normal circumstances is Vy flap 0 which is 100kias (is there a Vy flap 10?).
By the way, which is the equivalent of, or the closest speed to, best L/D ratio on aircraft like the Twin Otter?