According to Wikipedia, three spellings can be found for the Griffin, and the source of the word is uncertain. Is it simply a case of different sources choosing their favorite spelling that led to three different spellings?
Ok, remember this
Gryphons/Griffins/Griffons are originally greek. It also doesn't really have a literal translation, so we go by the phonetics.
I'll have to trust Wikipedia on this, but γρύφων is the greek word for the creature.
Also looking on the theoi page for this, most English translators agree that it is in fact Grypes. (Since you tagged it greek)
Aeschylus "[Prometheus warns the wandering Io :] ‘But now listen to another and a fearsome spectacle. Beware of the sharp-beaked hounds of Zeus that do not bark, the Grypes (Griffins), and the one-eyed (monôpoi) Arimaspoi (Arimaspians), mounted on horses, who dwell about the flood of Plouton's (Pluton's) stream that flows with gold. Do not approach them.’"
Aristophanes "Euripides : 'Twas all Skamandros (Scamander), moated camps, and Grypaietoi (Griffin-Eagles) flashing in burnished copper on the shields."
Ok fine. Grypaietoi. Same thing.
Herodotus "But in the north of Europe there is by far the most gold. In this matter again I cannot say with assurance how the gold is produced, but it is said that one-eyed men called Arimaspoi (Arimaspians) steal it from Grypes (Griffins). The most outlying lands, though, as they enclose and wholly surround all the rest of the world, are likely to"
Ctesias There is also gold [in India], not found in rivers and washed, as in the river Paktolos (Pactolus) [in Lydia], but in many large mountains which are inhabited by Grypes (Griffins). These are four-footed birds as large as a wolf, their legs and claws resembling those of a lion; their breast feathers are red, those of the rest of the body black. Although there is abundance of gold in the mountains, it is difficult to get it because of these birds." have those things which we think the finest and the rarest."
Pausnias "Grypas (Griffins), Aristeas of Prokonnesos says in his poem, fight for the gold with the Arimaspoi (Arimaspians) beyond the Issedones. The gold which the Grypas guard, he says, comes out of the earth; the Arimaspoi are men all born with one eye; Grypas are beasts like lions, but with the beak and wings of an eagle."
Note: There is an exerpt that he calles them Gryphoi.
"I have also heard that Gryphoi (Griffins) have spots like the leopard."
The rest of them are some variant of Grypes.
So, to go by one book (Guide to Greece) over many other sources is kinda unfair in my opinion, but to answer your question, its mostly just disagreeing on the phoenetics on the word Grypes. Or Gryphoi. They had not disagreed, mostly, on the spelling.