I have read somewhere that Shree Krishna is called the complete version of Vishnu compared to other of his 9 incarnations. Well it's understandable that Shree Krishna is different from Buddha, Narashima, Parashurama or even Vamana, the dwarf, but how is he different from Lord Rama? What qualities or powers did he possess that Shree Rama didn't?
Shri Krishna showed completeness in his Divine incarnation. He shows one quality in one event, and the exact opposite quality comes forth in another event. The abstract concept of Kalas is used to describe that Shri Krishna was a complete avatar:
Q. Gurudev, you have talked about Solah Kalas (referring to the sixteen extraordinary abilities or Divine qualities). Which are these and what is their importance?
Sri Sri Ravi Shankar: The word Kala usually refers to the extent by which the moon waxes (or wanes) in one day. So by this understanding, the Moon possesses 16 different Kalas (different phases of the Moon). You can observe this from one Amavasya (No-moon day) to the next Purnima (Full moon day). So if you count the number of days from one Amavasya to the next Purnima, it turns out to be 16. And the transition from No-Moon to the Full Moon also symbolizes moving from Zero (nothingness) to Fullness.
On the no-moon day you don’t see the moon at all, and on the full moon day you see the complete moon. This does not mean that the Moon is not there on the day of Amavasya. No, it is still there. So by this understanding, these are called the Solah Kalas. On the day of solar eclipse, for example, the Moon is there but it cannot be seen, that’s all.
So the word Kala is a unit of measure of one’s power, or by what degree one has blossomed or progressed. So, someone may be like a half-moon, someone else may have reached to three quarters of the moon and so on. This was a means of measurement in the ancient days...
...They say Lord Rama has 12 Kalas. Lord Hanuman is greater than Him in this regard and is said to have 14 Kalas. That is why he could help Lord Rama. Only a person who is stronger than us can help us...
...In the same way Lord Krishna is also said to have possessed all the Solah Kalas. Why is this so? In Lord Krishna, you see the complete and total manifestation of Divinity in every way. He is so complete. He is the King of thieves. He does not steal small things, he steals away your very mind. That is why He is called as Chita-Chora (the beautiful One who completely captures and steals away the mind).
He is also called as Ranchhodrai (meaning the cowardly one who flees the battlefield) in Gujarat. They say that He ran away from the battlefield in the middle of the war. But on the other side He lifted up the Sudarshan Chakra (the Lord’s divine discus) to defeat Bhishma Pitamaha (the grandsire of the Pandavas and the Kauravas in the Mahabharata) . So He is very brave and valiant too.
So, on one side he is very brave and on the other side he is like a coward. One cannot even imagine how complete he is. He is obedient as well, but He is also the one who breaks the rules.
In the Bhagawat, it is mentioned at many places that Lord Krishna is perfect and complete in every possible aspect. That is why it is said that one should not emulate Lord Krishna (Laughter). Follow the path of Lord Rama and listen to Lord Krishna. Don’t walk in the way that Lord Krishna did. If you go and steal the clothes of a lady, you will be in big trouble (jokingly referring to one of the pastimes of the Lord where He steals the clothes of some ladies bathing in the river and hides them on a tree).
Excerpts from: http://www.artofliving.org/wisdom-q-a-25-february-2014-qa-5
Lord Krishna being a complete Vishnu/Bhagavaan is mentioned in Srimad Bhagavatam(Bhaagvat Purana 1.3.28).
All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of the Lord, but Lord Śrī Kṛṣṇa is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incarnates to protect the theists.
Hence on the authority of Bhagavatam we accept krishna as complete Vishnu or as Supreme Personality of Godhead.
In His commentary on Bhagavad-Geeta(2.2) A.C. Bhaktivedanta Swami writes
The Sanskrit word bhagavān is explained by the great authority Parāśara Muni, the father of Vyāsadeva. The Supreme Personality who possesses all riches, all strength, all fame, all beauty, all knowledge and all renunciation is called Bhagavān. There are many persons who are very rich, very powerful, very beautiful, very famous, very learned and very much detached, but no one can claim that he possesses all riches, all strength, etc., entirely. Only Kṛṣṇa can claim this because He is the Supreme Personality of Godhead.
This definition is given in Vishnu Purana.
In Chapter 10 of Bhagvad Geeta Krishna shows His opulence; that He is full in these 6 opulence.
No other incarnation as long as I know exhibits such opulence; However I would like to mention that Krishna,Ram,Vishnu are considered to be expansions of same supreme Lord and are non-different from Gaudiya Vaishnava perspective. However for welfare of people, to uphold religion and anhilate the demons god has assumed different forms at different times.