Particles have natural levels of focus. For instance, が is very strong especially for questions because that’s what you’re trying to identify. Emphasis can also be changed by tone of course. But ordering also plays a subtle role.

For example
ここにあったケーキを誰が食べた？
誰がここにあったケーキを食べた？

While the main focus is on 誰が (the heart of the question), the emphasis on 誰が is slightly weakened in the second sentence due to the time lag to the main verb.

That is the only consistent difference in sentence ordering that I can think of. I suppose there’s also elements of style to consider. For instance, in most cases it’s probably better to introduce a new topic at the beginning of the sentence. But I think that’s more in the realm of common sense, style, and composition rather than language learning.

I would similarly guess that which argument is placed first in Japanese is dependent on this sort of consideration, which is called “information structure” in linguistics. I’m not saying I know exactly what factors Japanese people take into consideration when they chose to have 犬が first or 木を first. However, the choice cannot be random, and thus the sentences are different. This means that in Japanese, sentence order -does- matter.

Some people would contend that there is an “unmarked,” or “least special” way to use a transitive verb in Japanese (assuming that you leave in all of the nouns). That order would be SOV. In English, it is SVO. As to whether this is true or not, I’m not sure, but it’s disingenuous to say that any linguist claims that -the- Japanese sentence order is SOV. Evidence for this may come from subordinate clauses, which may more frequently prefer the SOV order, because information structure does not make demands on subordinate clauses – thus the unmarked order can emerge. It certainly happens in English:

I gave that toy to my cousin.
To my cousin, I gave that toy.
The person who I gave that toy to
*The person who that toy, I gave to.

Putting the object in front is fine in the main sentence (forming an OSVO sentence), but not okay within a relative clause. The Japanese example I gave above is similar.

犬が木を見た
木を犬が見た

Everything I said in the above post applies to these two sentences – they are different for the reasons I stated. You cannot just change the word order whenever you feel like it – different orders have different functions in a conversation.

And everything else in my comments still stants wrt Japanese word order.

The two sentences are totally different because you switched the particles not just the ordering of the words. Remember, the particles define the function of the word so they are the most important.

犬が木を見た
木が犬を見た

are not the same sentence – that is, they are used in different situations with different background assumptions (about who in the conversation is important, who is being talked about, and so on).

Also, in so-called subordinate clauses (aka noun-modifying clauses), word order is often more fixed. It sounds odd to me to have something like

俺のコンピュータをその人が修理した日

This sounds better: その人が俺のコンピュータを修理した日. At least, in an ‘out of the blue’ context, the first sentence sounds unnatural. You don’t have to say that Japanese is -always- SOV, which it’s not, just that SOV is the unmarked, non-special word order. It can be mutated according to conversational needs. Separate principles will let you omit the subject or object if you want to, but if you did want to say them, you’d have to know where to put them.

Please read the comments above for more clarification.

I’d rephrase it to just
“Myth: English sentences are SVO and Japanese sentences are SOV”. Maybe followed by “and smack your teacher if they keep claiming this” maybe >_>

However, in Japanese, teaching SOV causes nothing but trouble. I see it all the time, people saying 私は for every sentence about themselves. Or the confusion when teacher say using the polite word for “you” （あなた） is impolite. Students don’t realize that most sentences in coversations don’t even have a subject. So perhaps I should rephrase “myth” to “a horrible teaching method that should never be used but nevertheless is taught at countless number of places”.