Well, besides all of the inconveniences brought on by Alolan exeggutor’s long neck (Alolan exeggutor are also a pain to clean and feed because of this), there is the fact that Alolan exeggutor are dragons, whereas exeggutor from elsewhere are psychics. Alolan exeggutor behave like dragons as well, from their inability to withstand cold (exacerbated by the fact that they’re partially plants as well) to their naturally fiercer natures. On the positive side, while their weakness to bug-types is not completely negated, they gain more of a resistance to insects than their psychic counterparts, and fire is barely an issue at all thanks to their dragon element. In short, the most important thing to remember about an Alolan exeggutor (besides all of the problems associated with its neck) is the elemental shift. Types that would greatly harm an average exeggutor may not affect an Alolan exeggutor the same way—but types that might not affect an average exeggutor that much may aversely affect an Alolan.
Of course, there are other things to keep in mind. As I’ve briefly mentioned above, Alolan exeggutor behave like dragons as well, so they may be slightly more aggressive than their psychic counterparts. Doubly so because their psychic counterparts prefer standing in one place and blending in with their surrounding forests, whereas Alolan exeggutor, powered by the brighter sunlight of its native region, prefer to defend themselves with acts of aggression and dominance.
Also, be wary about the tail. While not as fragile as its neck, the bulk of the tail is difficult for an exeggutor to defend. However, like girafarig, the seed at the end is actually another head, and this head is capable of thinking and acting on its own to defend the rest of its body from behind and below. (It does not need to be fed a separate meal, but some trainers find it more convenient to let their exeggutor eat using it instead of its main heads.) In other words, don’t assume that just because your exeggutor has its back turned towards you that it’s defenseless. If anything, standing behind your exeggutor is more dangerous than standing in front of it. At least if you stood in front, it can only rely on its main heads to see you and attack (not that you should provoke your exeggutor in the first place).
Finally, Alolan exeggutor rely more on sunlight than their psychic counterparts and should be kept outdoors at all times (not that … you would find much of a place for them indoors). It is, after all, the brightness of the Alolan sun that allows them to grow to such a height, so not exposing them to constant, direct sunlight may make them lethargic or stunt the growth of a newly evolved exeggutor.
Good luck, @dragonlikeleaves. May this prove helpful to you as you, presumably, raise your own exeggutor.