i can't speak for crochet, but knitting didn't exist until the 9th century AD, its among the youngest of fibre crafts. until recent history it was mostly used for making garments such as hats and socks, due to it stretching in every direction (unlike woven fabric and unlike crochet, which doesn't stretch much at all) rather than its qualities of making warm garments. as to my knowledge (i only knit) knitting is no warmer than other fabrics of similar thickness. the fibre type used has a much greater influence of the warmth quality of the fabric.

you'll see a lot of people claiming that they've seen much more ancient pieces, but these are always other crafts, such as naalbindig (literally "thread binding") which possibly predates weaving.

i'm more than aware that i'm supposed to cite sources, but i don't have any. i'm just a knitter.

Crochet, as crotchet, a.k.a. "shepherd's knitting" became popular and common about 1800. I've been spending a lot of time in old books of needlecrafts and fancywork, and this is what authors of the early 1800s said about its history. While it may have been used here and there before then, you don't find it in common use. About this time, you find descriptions of how-to in English, French & German.

By the Middle Ages knitting was quite sophisticated. There's a Madonna & Child in Boucher's 20,000 Years of Fashion that shows a stocking being made in four-needle technique.