Hekmatullah Hekmat, a district governor in Herat, says both incidents took place yesterday.
The children were riding a motorcycle that struck the mine and blew up.
The acting governor in Herat, Juma Adil, says the two soldiers died trying to defuse the other mine. A third soldier was wounded in that explosion.
No group immediately claimed responsibility for the attacks.
Afghanistan has the highest number in the world of victims of land mines, which along with roadside bombs kill or wound an estimated 140 people every month.