Which two conflicts does Lear face during the storm in King Lear?
It is clear that the storm that dominates in Act III is symbolic of the turmoil that is engulfing Lear's kingdom and Lear himself. It appears that the two major conflicts Lear faces in Act III scene 2 are internal and external. His increasingly unfocused and bizarre comments suggest that he is facing an internal conflict to maintain hold of his sanity. Lear himself recognises that his "wits begin to turn" as he rages against his two ungrateful daughters that have cast him out.
Secondly, Lear obviously faces and comments upon the external conflict he has with his daughters, which has resulted in his exile status as he wanders through the countryside seeking shelter. Notice how Lear addresses the storm in regard to this external conflict:
Rumble they bellyful! Spit, fire! Spout, rain!
Nor rain, wind, thunder, fire, are my daughters.
I tax not you, you elements, with unkindness;
I never gave you kingdom, called you children.
You owe me no subscription.
Because the storm does not "owe" Lear anything, he encourages it to rage on, in contrast to his "two pernicious daughters," who owe their father a great deal but have done nothing to help him, and instead have spurned him, throwing him out of their houses.