Rama and Krishna avataras are considered complete avataras. Only in these two avataras did the Lord stay on this earth for many years, said Damal Ramakrishnan. When the Lord appeared as Narasimha, Prahlada begged Him to stay back at least for the night. But the Lord replied that once His task had been accomplished, He wouldn't remain on this earth. But He showed no such haste in leaving in the Rama and Krishna avataras. In the Rama avatara, the purpose of the avatara was the killing of Ravana. When Rama had killed Ravana, the celestials appeared before Him and said: “The purpose of your avatara has been fulfilled. So come back. You only have to come back to this earth later to slay Kamsa.” But the Lord replied that they should not expect Him to leave at their bidding. He was not going to end the avatara simply because they asked Him to. Rama lived for 11000 years on this earth.
Krishna too lived for many years. His purpose was to kill Kamsa, At least Rama was born in Ayodhya. So He had to wait to be banished by His father, and for Sita to be kidnapped before He could go to Lanka to kill Ravana. But in the case of Krishna, He was born in a prison, which was within the kingdom of Kamsa. So He could, if He had wished, have killed Kamsa right away and departed to Vaikuntha. But instead, He instructed His father Vasudeva to take Him to Gokul.
Thus He resorted to a very circuitous route to reach His goal of killing Kamsa. Why did He do this? He did this because of His love for us. He wanted to show the world His simplicity and His concern for us. That is why He stayed on.
Koorathazhvar said that while Rama had one set of parents belonging to one gotra and spent His childhood in one place, Krishna had two of everything: he had two sets of parents (His biological parents and His foster parents); and two gotras (that of Vasudeva and of Nanagopa). While Rama spent His entire childhood in Ayodhya, Krishna spent His in two places.
Narayana Bhattadri, in his Narayaneeyam, said Rama was a purna avatara, and Krishna was a purna punya avatara, thereby placing the Krishna avatara one step above the Rama avatara.