Tom is fairly clever, but Peter is rather stupid.

I walk fairly fast but Ann walks rather slowly.

Both can be used similarly with participles:

He was fairly relaxed; she was rather tense.

a fairly interesting film

a rather boring book

The indefinite article, if required, precedes fairly but can come before or after rather:

a fairly light box      a rather heavy box/rather a heavy box

With adjectives/adverbs such as fast, slow, thin, thick, hot, cold etc., which are not in themselves either ‘favorable’ or ‘unfavorable’, the speaker can express approval by using fairly and disapproval by using

rather: This soup is fairly hot implies that the speaker likes hot soup, while This soup is rather hot implies that it is a little too hot for him.

B.     rather can be used before alike, like, similar, different etc. and before comparatives. It then means ‘a little’ or ‘slightly’:

Siamese cats are rather like dogs in some ways.

The weather was rather worse than I had expected.

rather a is possible with certain nouns: disappointment, disadvantage, nuisance, pity, shame and sometimes joke:

It’s rather a nuisance (= a little inconvenient) that we can’t park here.

It’s rather a shame (= a little unfair) that he has to work on Sundays.

fairly cannot be used in these ways.

C.     rather can be used before certain ‘favorable’ adjectives/adverbs such as amusing, clever, good, pretty, well but its meaning then changes; it becomes nearly equivalent to very, and the idea of disapproval vanishes:

She is rather clever is nearly the same as She is very clever.

rather used in this way is obviously much more complimentary than fairly.

For example the expression It is a fairly good play would, if anything, discourage others from going to see it. But It is rather a good play is definitely a recommendation. Occasionally rather used in this way conveys the idea of surprise:

I suppose the house was filthy. ~ No, as a matter of fact it was rather clean.

D.   rather can also be used before enjoy, like and sometimes before dislike, object and some similar verbs:

I rather like the smell of petrol. He rather enjoys queuing.

rather can be used in short answers to questions with the above verbs:

Do you like him? ~ Yes I do, rather.

rather + like/enjoy is often used to express a liking which is a surprise to others or to the speaker himself. But it can also be used to strengthen the verb:

I rather like Tom implies more interest than I like Tom.