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.
Probably we should not look to much at what a dictionary says, but more at what is used in practice. If a certain way to say things is common, hw can a dictionary say it is wrong? Besides, I think there are regional differences in the way English is spoken. At least this holds for Dutch, which is a much smaller language in number of speakers. Except for a small group of purists, Dutch language experts tend to accept anything that is common.
I agree that it's what is commonly said that counts. Actually, dictionaries are very good at describing what people say rather than prescribing what they should say, (hence the usage notes at the dictionary link I gave above). It's those purists you have to watch out for :-)
However, we sometimes have to be more formal in a language course just to show that we have understood the grammar. In the case of this question, "Mnahitaji nguo ambazo zina joto", although (like Kola871084) I would actually say "You need warm clothes", I may have to follow Joel158879's advice to use the relative pronoun "which" or "that" just to show that I understood the Swahili word "ambazo".