One of the best-known differences between British and American English is that BrEng has words ending in -our where AmEng has -or. It is not an absolute rule that BrEng nouns have the u; rigour is used in the general sense but, following Latin directly, there is rigor (mortis) in medical usage. On the face of it there seems to be no reason why horror and honour should differ or why error should not be errour.
The OED has an entry for honour, honor
[a. OF. onor, -ur, honor, -ur (11th c.), AF. (h)onour, mod.F. honneur (= It. onore, Sp., Pg. honor):—L. honōr-em repute, esteem, official dignity, honorary gift, ornament, grace, beauty.
…. Honor and honour continued to be equally frequent down to the 17th c. In the Shakespeare Folio of 1623 honor is about twice as frequent as honour. The two forms appear indiscriminately in the early 17th c. dictionaries, but honour was favoured by Phillips, Kersey, Bailey, Johnson. Ash, 1775, adopted ‘Honor (a modern but correct spelling)’, and this is said to have been fashionable at the time (see quots.). Nevertheless honour carried the majority of English suffrages eventually, while honor was (under the lead of Noah Webster) generally accepted in U.S. As to derivatives, Bailey, 1731, considered honorable, honorary, ‘the best* spelling’, but referred them to honourable, honourary, as the more usual. Phillips, in his various edd., had honorary, Kersey (1706) honourary or honorary, Chambers (1727–41) honourary. Johnson, 1755, has honour, honourable, honorary.
*On a completely different matter, it is worth noting this use of best rather than better with reference to two possibilities.
For the etymology of horror it merely says without explanation:
[a. OF. orror, (h)orrour (mod.F. horreur) = Pr. and Sp. horror, It. orrore:—L. horrōr-em, f. horrēre to bristle, shudder, etc.
Its most recent quotation for horrour dates from 1756:
On the return of reason he began to conceive a horrour suitable to the guilt of such a murder.
For error it explains:
Down to the end of the 18th c. the prevailing form was errour, which is the form given by Johnson and by Todd (1818); Bailey's Dict. introduces error in 1753, and this spelling is now universal. (In words which have -rr- before the suffix, as horror, terror, mirror, the spelling of -or for an older -our is accepted by British as well as American writers.)]
That last point about -rr- words is one of those wonderful English rules that are so obscure that it is easier, if necessary, to learn the exceptions.
It is, however, standard that AmEng has -or but even there, there is one exception: glamour with the u is more common than glamor. The only explanation that I have found for this, from the Mighty Red Pen, is that ‘glamor just doesn’t seem right’.
According to Google Ngrams, in AmEng glamour beats glamor (click to enlarge the images):
but in BrEng errour barely survived beyond 1830: