What is the difference between Apollinarianism and Eutychianism? Both Apollinaris and Eutychius believed in one divine subject Christology that the only and whole person of Christ is a divine person, the Logos. Not a divine-humane person. Their Christology were condemned at Constantinople (381) and Chalcedon (451), respectively. Apollinaris teaches that the Logos supplant the rational faculty of Christ not another man. While Eutychius teaches that the flesh of Christ belongs to the Logos and not another man. So that both teach the Logos, a divine person is the only divine subject. How then one in principle manner differentiate their Christology from one another?
In general, the Apollinarians said that Christ's humanity is incomplete because He has the human body and human soul, but does not have the human spirit, which is mysteriously replaced by the Logos. The so-called Christ is thus formed with these three: Logos, soul, and body (unlike a normal human being which is made up of the spirit, soul, and body). 

According to Eutychians, they denied the distinctness and coexistence of Christ's divinity and humanity, but asserted that the two natures were merged into one, which is neither divine nor human, but a third nature resulting through the merging of divinity and humanity - the divinity being dominant, with the humanity being absorbed by the divinity. Hence the Eutychians were also known as the Monophysites.

Eutyches did not seem to reject the full humanity of Jesus Christ. But, even though he did not repeat the exact heresy of Apollinarius, he did reduce Christ's humanity to a "drop of wine in the ocean of his deity".

Justo González, in "A History of Christian Thought, vol. 1", mentions that: 


  ... Eutyches, a monk in Constantinople who lacked theological subtlety,
  and who held that, while the Savior was “of one substance [homoousios]
  with the Father, he was not “of one substance with us.


According to Eutyches, the humanity of Jesus Christ made no different to the the Logos and was swallowed up in the incarnation union with Him. So, not only Christ did not have a human personality or individual human existence, he did not have a human nature like ours at all.

Basically, a third nature in which neither human nor divine but in which the divine dominates.

So, the difference is Apollinarians states that Christ's humanity is incomplete while Eutychians merges both natures into a third, losing the distinctness and co-existence of both.