Let me borrow from the Stanford Encyclopedia of Philosophy to briefly highlight the fundamental differences between the two:
Broadly speaking, there are, at bottom, only two positions that can promise the desired integration: panpsychism and emergentism. If one believes that the most fundamental physical entities (quarks, leptons, bosons, or whatever physics will ultimately settle upon) are devoid of any mental attributes, and if one also believes that some systems of these entities, such as human brains, do possess mental attributes, one is espousing some kind of doctrine of the emergence of mind. All the currently popular physicalist theories (such as behaviorism, central state identity theories, functionalism) are theories which attempt to provide an account of how the mental emerges from the physical. Other, more radical, forms of emergentism are possible. These are theories that deny that there is any explanatory account of how emergence works: it is a brute fact of natural law that certain configurations of physical entities underpin certain mental states. This fact can only be accepted, as Samuel Alexander said, with “natural piety”. Of course, an inexplicable or brute emergence is still a form of emergence. It is possible to mark the distinction between the forms of emergence in terms of the explanatory role of the mental. Modern materialists do not regard the emergent features of the mind as either ontologically or explanatorily fundamental. The more radical emergentists would regard mind as explanatorily fundamental, but not ontologically basic in the sense that material conditions (or in general some system of non-mentalistic conditions) are required for the existence of mental features. The panpsychist naturally regards mind as both explanatorily and ontologically fundamental in the sense just mentioned.
Now, in the Awakening of Faith in the Mahayana the following question is proposed.
If the dharmakāya of the Buddhas is separate from the characteristic of form, how is it able to manifest the characteristic of form?
The answer is then provided:
答曰：「即此法身是色體故，能現於色。所謂從本已來色心不二，以色性即智故色體無形，說名智身；以智性即色故，說名法身遍一切處。所現之色無有分齊，隨心能示十方世界，無量菩薩無量報身，無量莊嚴各各差別，皆無分齊而不相妨。此非心識分別能知，以真如自在用義故。」」(CBETA, T32, no. 1666, p. 579, c11-19)
As the dharmakāya is the essence of matter, it can manifest as form. It is said that from the beginning matter and mind are non-dual. As the nature of matter is wisdom, the essence of matter is without shape and this is called the wisdom body. As the nature of wisdom is matter, the dharmakāya is said to pervade all places. Manifested matter has no limit. According to mind [freely] it can reveal the worlds of the ten directions, immeasurable Bodhisattvas, immeasurable saṃbhoga-kāya, immeasurable ornaments all individually distinguished and all without limit not mutually obstructing one another. This is not something the [unenlightened] mind or consciousness distinguishes or can know because it is the meaning of the unimpeded function of suchness.
Basically, what came to my mind was that from the perspective of panpsychism the idea of the dharmakāya, which essentially is an omnipresent and omniscient awareness, is not at all unrealistic and it even makes sense. A Buddha is a being which takes the universe, which is pervasive with matter that is essentially aware, as its body and this would support the case in general terms for the omniscience of said entity. Moreover, that as components in the totality of the universe all sentient beings would be encompassed within the womb of that being which has taken the entire universe as its body. This would in turn explain the unconditional compassion of a Buddha. A Buddha essentially feels and knows it is all beings simultaneously.
If all matter is aware to some degree, then what prevents me from being aware of only the matter which is my body and what I can sense within my immediate area? From one Buddhist perspective the answer is that defilements prevent omniscience and once these defilements are entirely removed there are no more hindrances to it. A Buddha is a being which has entirely removed all defilements ergo it is omniscient.
Now, as for the evidence for what I've proposed here the text also states that only high level Bodhisattvas can actually actively perceive the dharmakāya while lesser Bodhisattvas can have glimpses of it. In other words, this is not at all verifiable through scientific observation. However, that being said, panpsychism itself can be the lens through which science is conducted. What I'm saying here is that if we assume all matter is aware, then why am I only aware of a limited amount of it? If it is because of some kind of obstacle which can be removed such as defilements as defined by general Buddhist doctrine, then could my awareness come to encompass all matter?
I wonder if we can utilize panpsychism and integrate it into the explanatory framework for how a purportedly omniscient being could come to exist?