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?
The orthodox (Chalcedonian) position is that Jesus is fully God and fully man, maintaining the distinction of the natures but also the unity of the natures. Within the human nature, theologians often recognize three components: a human mind, human body, and human soul.
Apollinarians denied Jesus' full humanity. They affirmed his full divinity, and his human soul and body. But they denied he had a human mind, affirming that Christ's mind was the Logos.
Eutychianism denied the distinction of Jesus' natures. They affirmed his full divinity and full humanity, but believed that the two natures merged into one so that there is no distinction at all. Eutyches was apparently not much of a theologian, so exactly what this meant originally is not entirely clear. But he was clearly influenced by the Christology of Cyril, who stressed the unity of Christ's natures to the point that later followers of his (the Oriental Orthodox of today) deny that Christ's natures remain distinct.
For example, we may say that the person of Christ suffered on the cross. But since suffering is something enabled by human nature and God cannot suffer, we may also say that Christ suffered according to his human nature (and not according to his divine nature). That would be Chalcedonian. The Oriental Orthodox reject that distinction. They would say, whatever happened to the person of Christ happened because of his one nature.
Eutyches may not have thought his Christology through, but those who reject the council that condemned his Christology certainly have. There is debate about whether the "miaphysitism" of the Oriental Orthodox can properly be identified with the monophysitism of Eutyches, but it seems the Council of Chalcedon had the same effect on both: putting them out of the church.
Summary: Apollinarians denied Jesus' human mind. Eutychians denied that Jesus' humanity and divinity were separate.
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.