In British English, when words have an '-our' suffix (eg favour, humour & vapour), get another suffix (eg '-able', '-ous' & '-ise' ), they lose the 'u' (eg favorable, humorous & vaporise). This is a standard rule, therefore there isn't the word 'favourable' in British or American English.

There is a similar rule with the suffix '-ous': Curious ----> Curiosity But in this case, pronunciation changes as well. This applies for both British and American English.

My dictionary, Gyldendal's Red Dictionary translates from Danish to British English and vice versa and is the most reliable one in Denmark (which is very reliable, even though Denmark is a small country).

It knows the word "favourable" and not the word "favorable". And it spells "vaporize" as so, just like the American spelling. Cambridge on the other hand has results for both "favourable" and "favorable", but neither one has "vaporise", Cambridge even redirects to "vaporize" when searching for it.

Also my British spell checking plug-in for Firefox agrees on "favourable", "vaporize" and is even cool with "vaporise", but it puts a red line under "favorable". This plug-in might not be that reliable, though, I don't even know who made it.

British English spelling is — or has been — more variable than one might imagine. (Microsoft Word’s spelling dictionary and its ilk seem currently to be forcing a standardization — or ‘standardisation’, as it would have me write.) Using individually compiled printed dictionaries as reference, here are some relevant extracts from Chambers and the Oxford English Dictionary (OED).

Chambers (contemporary iPhone version and printed 1993 ed.)

‘vaporize’ or ‘vaporise’

and

‘favourable’ or ‘favorable’ (esp US)

OED (encyclopedic edition 1991)

‘vaporize’

and

‘favourable’ (US ‘favorable’)

OED (1926)

‘vaporize’ (18thC citations also include ‘vaporise-’), but also ‘vapourized’, citing usage by Macaulay as equivalent to ‘vapoured’, having the different meaning, ‘filled with vapour’.

and

‘favourable’, ‘favorable’

This edition (unlike the later) gives both spellings, ‘favour’ and ‘favor’, with the note: “As with other words of the same ending, the spelling with -our is preferred in the British Isles, while in the US -or is more common.” In its examples the latest use of the spelling ‘favorable’ is by Hobbes in 1651 (although there is one example of ‘favorably’ from 1872).

(N.B. ‘-ise’ seems only to have become more frequent than ‘-ize’ in British English usage in the latter half of the twentieth century. The OED (1928), for example, includes ‘recognize’ but not ‘recognise’; and this is still the case for a pocket edition I have from the ’50s.)

"Vapourise" or "vapourize" does seem to be considered an incorrect spelling of this word in any form of English.

To understand why, it helps to know a little bit about the history of words that are spelled with ⟨our⟩ in British English.

⟨our⟩ comes from a French version of Latin ⟨or⟩

Many Latinate words came into English from French (by which I mean, not modern French, but the Anglo-Norman dialect that was spoken by French speakers living in Great Britain). The British English practice of using the spelling ⟨our⟩ in some words where Americans use ⟨or⟩ usually reflects the phonetic development of Latin "or" to French "our" (in Modern French, as Peter Shor mentions in a comment, the "ou" has often changed further to "eu"). Therefore, in words that are taken directly from Latin (or at least, treated as if they were taken directly from Latin), without the intermediary of French, we would generally expect to see ⟨or⟩ instead of ⟨our⟩.

The Latin spelling ⟨or⟩ occurs before suffixes that strongly prefer to come after Latin roots (-⁠ation, -⁠ious)

Certain suffixes, such as -(a)-(t)-ion, "prefer" to come after the Latin form of a root. As an example, the verb destroy, which has a form showing some French sound changes, cannot be used unchanged as the base of an -(a)-(t)-ion noun: we don't say or write *"destroy-tion" or *"destroy-ation". Instead, we have the noun destruction, which contains the Latin spelling of the same root.

Another pair like this is deceive, deception.

So, the spelling difference between colour, coloration is parallel to an audible difference that exists for some other sets of words. Hopefully this is helpful for remembering it.

The suffix -ious also generally selects the Latin spelling ⟨or⟩, which explains labour vs. laborious.

⟨our⟩ remains unchanged before suffixes that strongly prefer to come after whole words

Certain other suffixes are generally attached to complete English words of any origin. These suffixes attach to Latin roots only when the roots exist as independent words in English.

Inflectional suffixes take ⟨our⟩: -⁠ed, -⁠ing

All inflectional suffixes fall in this category. Inflectional suffixes for verbs include -ed, -(e)s, -ing; so we write the inflected forms of destroy as destroyed, destroyed, destroying, and of deceive as deceived, deceives, deceiving. Destructed does exist, but not as a form of deceive: it's the past tense of a different verb, to destruct. And *decepted does not exist; it would only exist if there was a verb "to decept"). Likewise, ⟨colored⟩ would only make sense as the past tense of a verb to ⟨color⟩, which does not exist in the type of British English you're talking about.

Some derivational suffixes take ⟨our⟩: -⁠hood, -⁠ship, -⁠ment

A number of derivational suffixes also fall into this category. Many of these are inherited or native suffixes, such as -hood, -ship and -ful: there is no change in pronunciation or spelling between neighbour and neighbourhood, saviour and saviourhood (or saviourship), or colour and colourful. This category also includes some non-native suffixes, though, such as the derivational suffix -ment (from French), which generally attaches to a stand-alone verb, not a bare Latin root (so Collins gives discolourment, from to discolour, and marks discolorment as an American spelling).

Fowler (1926; republished 2009) also mentions -ist and -ite as suffixes in this category. Labourist, behaviourist, colourist all prefer ⟨our⟩ in British English. However, florist does use Latin ⟨or⟩, as does arborist, from Latin arbor. Interestingly, humorist seems to be more common than humourist in British English (although both exist); see the Google Ngram Viewer and the British National Corpus (which has "humorist" with a frequency of 15, vs. "humourist" with a frequency of 4). I don't think -ite is really common enough to be worth worrying about, but the example Fowler gives is "Labourite".

Words that are standardly spelled with word-final ⟨or⟩ in British English unsurprisingly keep the ⟨or⟩ unchanged when suffixed, forming words ending in ⟨orist⟩: motorist < motor, terrorist < terror.

Either ⟨or⟩ or ⟨our⟩ seems possible in theory before the suffix -able, but the standard spelling is currently ⟨ourable⟩

Some other suffixes, such as -able, behave both ways, and can come after either a whole word of any origin, or a Latinate root. This suffix did exist in Latin, and it can be used in English after Latin roots (although for some words the spelling may change to -ible) as in destructible. However, -able is also highly productive after regular English verbs of any origin, so we also can say destroyable (even if some people might consider it less elegant). In some cases, we don't even use the word formed from the Latin root, only the one formed from an English verb (we say deceivable, not deceptible). Because of this, I think that both favourable and favorable are theoretically possible forms: to find out which is actually used more, it's necessary to consult a dictionary or corpus of British English.

Fowler advises using ⟨our⟩ before -able, and this is backed up by a look at the Google Ngram Viewer for "colorable" vs. "colourable" in the British English corpus.

⟨or⟩ is usual before the suffix -ise/-ize, which prefers to come after Latin roots (although it can also come after English words)

The suffix -ize/-ise is Latinate, and often prefers to attach to a Latin root (we say anglicize, not *englishize) but in modern English there are also many words formed by attaching -ize to whole English words (the Oxford English Dictionary lists foreignize, jeopardize, womanize among others). I think this is what leads to variation in the spelling of words such as colorise/colourise and vaporise/vapourise. There's actually similar variation with the words crystallize/crystalize and metallize/metalize (where the Latin stems, seen in crystalline and metallic, had double ll, but the independent English nouns crystal and metal have single l).

Dictionaries rarely list vapourise/vapourize: it is not usual and is likely to be considered incorrect

Tristan r left the following useful comment referencing several British dictionaries that have entries for "vaporise":

the Cambridge English Dictionary says (UK usually vaporise). See dictionary.cambridge.org/dictionary/british/vaporize?q=vaporize The Longman English Dictionary says the same ldoceonline.com/dictionary/vaporize

I did not find any dictionaries indexed by OneLook Dictionary Search that have an entry for "vapourise", aside from Wiktionary, which calls it a "Misspelling of vaporise."

Based on this, I would guess that many British English writers would indeed consider it a misspelling.

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The following question is also quite relevant: Should "glamourous" be considered incorrect?