'that' is restrictive. The information it heads is required:
A suitcase that doesn't have a handle is useless.
If we omit the underlined portion, the sentence is strange:
A suitcase is useless. :shock:
'which' is non-restrictive. The information it heads is not required:
A suitcase without a handle, which is old and ugly, is useless.
Place a comma (,) before and after the non-restrictive clause to show that the information isn't necessary.
'that' is often omitted when it functions as a conjunction: when it introduces a clause:
I like book that you bought.
I like the book you bought. (OK. 'that' is not required)
'you bought' is a clause: it has a subject ('you') and a verb ('bought'). Let's now look a sentence wherein 'that' heads a clause. That is, 'that' functions as the subject:
A suitcase that doesn't have a handle is useless.
A suitcase doesn't have a handle is useless. (Ungrammatical. 'that' is required)
'doesn't have a handle' is not a clause. It lacks a subject. In fact 'that' functions as the subject, so 'that' can't be omitted.
Let's now change the sentence so that we have both 'that' and a subject:
A suitcase that you gave me is useless.
A suitcase you gave me is useless. (OK)
'you gave me' has a subject ('you'), so 'that' can be omitted.
