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Krishna and Rama are both vishnu-tattva, in the category of the Supreme Person. But one difference is that Krishna is the origin of all other incarnations of God, and Lord Ramachandra could be said to be one aspect of Krishna's personality. Lord Rama's pastimes are those of a valiant king, and His superhuman activities are those of a great hero.
Krishna also performed incomparable feats in his killing of demons in battle when He was acting as Prince of Dwaraka. But His pastimes with His devotees in Vrindavan show an even more private side of His personality, without any trace of the majesty associated with His pastimes in Dwaraka or those He displayed as Ramachandra.
Krishna's Vrindavan devotees have such pure love for Him that they treat Him as an equal—either as a friend, lover, or son. It is said that in Dwaraka, where Krishna appears as royalty, His majesty overpowers His sweetness. But in Vrindavan, where Krishna appears as a simple cowherd boy, His sweetness overpowers His majesty.
As Ravindra-svarupa dasa (William H. Deadwyler, Ph.D.) mentions in the ISKCON Communications Journal, #2.1, January/June 1994:
"According to Gaudiya Vaishnava theology, God has both a public and a private face. When He manifests his power and majesty (aisvarya), He is known as Narayana and is served in awe and reverence. When He sets aside His lordship, however, and allows his beauty and sweetness (madhurya) to overpower his majesty, He is known as Krishna, the all-attractive. In order to enjoy intimate exchanges of love, Krishna causes His confidential devotees to forget that He is God, so that they may serve Him in a fraternal, parental or conjugal mood." Any avatar or embodiment came into being to play a role in history, to tell a profound tale. The two most unique avatars of Vishnu are Rama and Krishna. Both were symbolic of Godliness but couldn’t be more distinct.
When comparing the personalities of Rama and Krishna, the first thing that comes to mind is that Rama was still mature, stern, and straightforward. During his lifetime, he listened quietly to the troubles of the people, followed the counsel of his elders and did what was good or what was Dharma. No surprise, he was called the perfect man Maryada Purushottam. Krishna portrayed the playful side of Dharma while still following it. When telling others a lesson, he did not necessarily take the obvious route but instead saw the ironic side of it and used mischief, even trickery, to outsmart the villains and see that the weak were treated fairly.
When Ram, the Ayodhya Prince, was on the throne, he ruled his kingdom according to the dharma niti laws. The code was more important than his own judgement or instinct in determining what was right or wrong for him.
Krishna, on the other hand, ignored decorum and laws in order to provide immediate remedies, whether it was to save damsels in distress like Draupadi or to provide riches for his childhood friend Sudama. In a moment, he gained a spot in people’s hearts!