The figure holds together two things that sit oddly beside one another, and different Hindu traditions weight them very differently.
The Gita Krishna is a teacher of duty. He tells Arjuna, paralysed at the sight of relatives in the opposing army, to fight, and the argument he gives is not that killing is fine but that the self is not killed, that action performed without attachment to its fruit does not bind, and that Arjuna's own duty as a warrior is not evaded by refusing it. He then reveals himself in a terrifying universal form, all-consuming time. This Krishna has been read by Shankara, Ramanuja and Madhva to opposite metaphysical conclusions, and in the twentieth century by Gandhi as an allegory of inner struggle and by others as a straightforward endorsement of righteous war.
The Vrindavan Krishna is a child stealing butter and a young man playing the flute while married women leave their houses at night to dance with him. The theological reading, developed at length in Gaudiya Vaishnavism, is that the gopis' abandonment of social propriety is the model of a love that gives up everything for God, and that the intensity of the metaphor is the point. The devotional literature is explicit that this is not licence.
Colonial-era European commentary found the second Krishna scandalous and said so, and a strand of nineteenth-century Hindu reform responded by emphasising the Gita and playing down Vrindavan. That argument shaped which Krishna got exported.