Daphnaeus and Daphnephoros both are appropriate epithets of Apollon. Daphnaeus refers to Apollon as the god who is of the laurel. This connects the laurel intimately with his identity as much as the bow and the kithara or lyre. In a mythic context we can imagine Apollon bringing the laurel into his identity as he wreathes his head with laurel leaves as his sacred plant. Artemis is also called Daphnaea which draws a similar association between laurel and the plant, but with like with so many cases where she shares an epithet with her twin it is unclear if one is borrowing the epithet from the other, and if so which god had it first? It has been speculated, by Pausanias I believe, but I might be misremembering which ancient author I got it from, that Artemis was called this because her image was made from wood of the laurel. It is quite plausible if that is a potential reason for her possessing that name that Apollon too may have had cult images made of laurel wood at some time. We do know that according to Delphic myth that the first temple of Apollon at Delphi was believed to have been made from laurel boughs. In such a case we can understand the laurel as something which Apollon is honored through and within. The presence of the laurel would thus be a vessel that represented the divine presence of the god. This seems reasonable as the Pythia was often pictured holding a bough of laurel either as a representation of the god, or perhaps her own identification with Daphne, the first priestess of Delphi and the first whom Apollon had loved.
Daphne herself is an interesting figure, especially if we look at the myths of Daphne with the Peloponnesian versions in mind. The Peloponnesians expanded upon the myth of Daphne to give her characteristics quite like that of Artemis, running with her own band of nymphs. Now this idea of a love of Apollon carrying characteristics of his twin is nothing new as we also see his wife Kyrene to be something of a reflection of Artemis as well. Before she had ever taken the form of the tree she was loved by a youth who disguised himself as a maiden because he loved her. Apollon, jealous, caused it to be revealed that the youth was male in which case he was expelled from their company. This then can be followed by the Delphic myth in which Apollon himself chases after the maiden who, seeing that she cannot outrun Apollon (and really who can? This is the god who can outrun Hermes) calls for help. In some versions her father, the river god Peneios (her father in Peloponnese is another important river..the Ladon) transforms her into a laurel tree just as Apollon embraces her. In another version she is swallowed by the earth and the laurel grows out from her tomb. It is likely this version which is referred to at the Daphne temple of Apollon where the god was depicted giving libation to the earth.
Speaking of the temple, the Daphne temple was located just outside of Athens and was an important stop for initiates on their way to Eleusis just before they went down to the sea just beyond the temple to purify themselves. The association with the temple specifically with purification is likely pertaining to the nature of the laurel in relationship to Apollon. He is the purifier and as Daphnaeus he is a god who purifies as the plant does. This is something we see in particular in vase paintings of the purification of Orestes in which the god stands behind the seated Orestes and a pig is slain above him for his blood guilt and nearby a laurel. In myth the laurel grew after the event to mark the spot where the god purified Orestes. Furthermore, in the reenactments of his purification following the Stepteria festival shows the god traveling down to the Temple valley where the Peneios river flows and there a sacred laurel tree provided branches to be cut for purification and carried back to Delphi. This tree may have been cultically regarded as the very tree that the nymph transformed into by the Delphinians.
The youth thus carries back the laurel, himself representative of the returning god just as he was representative of the slaying and fleeing god during the Stepteria, and is called as the Daphnephoros. He is the laurel bearer. Apollon himself acts as divine laurel bearer and those who carry the laurel in his honor are also called Daphnephoros. In Thebes there was a festival which I strongly suspect due to its nature and falling around the same time, was a parallel to this return of Apollon. The Daphnephoria was a festival in which Apollon, by oracle, indicated the means to stop a terrible ongoing battle. After which the matter being resolved a festival was initiated in which at the fore of the procession was a youth carrying laurel to the temple in the very manner in which we see at Delphi. Apollon comes to his temple bearing the items of purification with him. It was naturally considered to be of high honor to be the Daphnephoros (and as such came with strict requirements as did the position at Delphi) for the youth selected. Parents of the youth at the Thebes dedicated tripods to Apollon, including, according to Pausanias, the father of Herakles when he himself took the position in his youth, as Pausanias notes a tripod dedicated by his mortal father Amphitryon.
Apollon is thus the god of the laurel, but also the god who bears the laurel in wielding its purify influence, in which case Daphnaeus is descriptive of the god himself, and Daphnephoros is descriptive of an action of the god (much like the epithet Pythios as he himself isn’t rotting but causes the action of rotting), one which is symbolically carried out in festivals by the performance of youths acting his part in capacity of bringing forth the laurel in sacred events.