If I mix the order when using chigau, let's say I want to say my books are different than your's... If I mix up the order and it comes out as: your books are different than mine... does it really change the meaning?
Example 1: watashino hon wa anatano hon to chigaimasu Example 2: anatano hon wa watashino hon to chigaimasu
Basically your example sentences mean the same. However, the focus of the sentence changes a little depending on the topic of the sentence. What I mean is:
In the sentence (1), WATASHI NO HON (=my book) is the topic of the sentence, so it has more emphasis than ANATA NO HON (=your book). It is most likely that WATASHI NO HON already was the topic of the conversation.
In the example (2) on the other hand, ANATA NO HON (=your book) is the focus of the sentence.
Please look at the following casual conversation between older sister (A) and younger brother. You will see, in the last sentence, why WATASHI NO HON as a topic of the sentence is more natural:
A: わたしの にほんごの ほんを みなかった？(WATASHI NO NIHONGO NO HON O MINAKATTA? = You didn't see my Japanese book, did you?) B: これ？(KORE? = This?) A: これは、あなたの ほんでしょう？ しみが ついてるし やぶれてる。(KORE WA ANATA NO HON DESHOU? SHIMI GA TSUITERU SHI, YABURETERU = This is your book, isn't it? It has stains and it's torn.) B: えっ？あ、そうか。 (E? A, SOU KA = What? Oh, I see.) A: わたしのほんは あなたのほんと ちがいます！しみもついてないし、やぶれてない！(WATASHI NO HON WA ANATA NO HON TO CHIGAIMASU. SHIMI MO TSUITENAI SHI, YABURETENAI! = My book is different from yours. It doesn't have any stains and it's not torn either.)