In "differently coloured", coloured appears to be the past participle of a verb: socks that have been coloured in different ways.
In "different-coloured", we're attributing a type of colour to the socks. Compare "two-faced" or "bow-legged" or "flat-footed". A flat-footed person has flat feet, and different-coloured socks have different colours.
Hence, we're much more likely to say "different-coloured", because people don't colour their socks, but the socks' colours are different.
Also note Michael's comment about "odd socks". Here in North America we would say "mismatched socks" or change the structure entirely and say "your socks don't match". When asked to clarify, we might say "different-coloured socks". However, socks can also have different patterns and designs and the phrase "different-coloured" wouldn't be used if the socks differed in some respect other than colour.
A note on the hyphen. While punctuation is pure convention and not really a linguistic system, the convention is to use a hyphen for compound adjectives and no hyphen for a series of adjectives. I've used it above because "different" modifies "coloured". Whereas:
— Do you like the taste of bottled water?
— There are lots of different bottled waters. I like some and dislike others.
Here the lack of a hyphen after "different" allows us to interpret "bottled water" as a unit and to say that there are lots of different types of this category.