Many writers (more in Britain than the USA) distinguish between "that" and "which". "That" is appropriate in the sentence above: "You need clothes that are warm." See H. W. Fowler, Modern English Usage, lemma "that".
In British English, it really depends how formal you want to be. I am British and I do make this distinction (partly because US English and Microsoft Word are so strict about it), but it isn't incorrect to use "which" here (a restrictive relative clause).
According to the Oxford dictionary's usage notes on "that":
Is there any difference between the use of "that" and "which" in sentences such as "any book that gets children reading is worth having", and "any book which gets children reading is worth having"?
The general rule in British English is that, in restrictive relative clauses, where the relative clause serves to define or restrict the reference to the particular one described, "which" can replace "that".
However, "that" would definitely be incorrect in non-restrictive relative clauses (where the relative clause serves only to give additional information). Here, "which" has to be used, in both British and US English: "This book, which is set in the last century, is very popular with teenagers."
The difference is so subtle that people find it simplest to always use "which" in non-restrictive relative clauses" and "that" in restrictive relative clauses. Then they just need to remember that "which" clauses are set off by commas and "that" clauses are not.