The first example is a echoing a verb.

The verb being echoed is "spends." So you should read the second sentence in the first example as "The US spends more money on health care than Canada spends." The word "does" echoes the word "spends" and sounds better than actually repeating "spends."

On your second example, you are comparing nouns. So you simply state each noun with nothing implied. "This performance is more important than any other performance."

But what is the rule when do I have to add a word and when not??

I don't think that in example 1 there is any ambiguity, if we had omitted the word "does"?

Actually both the sentences that you have marked "wrong" in your first post and absolutely correct.

The "rule" is that only when there is an ambiguity, that is when you need to add the verb.

For example (an official example):

Incorrect:
Inuits of the Bering Sea were isolated from contact with Europeans longer than Aleuts or Inuits of the North Pacific and northern Alaska.

The reason this is incorrect is because it could mean two things:
x) the Bering Sea Inuits were isolated from Europeans longer than they were isolated from Aleuts and other Inuits or
y) the Bering Sea Inuits were isolated from Europeans longer than Aleuts and other Inuits were isolated from Europeans.

Correct:
Inuits of the Bering Sea were isolated from contact with Europeans longer than were Aleuts or Inuits of the North Pacific and northern Alaska. [/u]

confused13 wrote: Could someone explain me, when I have to add words and when not?

Rule 1: Clarity
The comparison must be crystal clear.
If more then one interpretation is reasonable, then the comparison must be clarified.
Generally, adding just a word or two will make the comparison crystal clear.

Rule 2: Parallelism
Generally, we compare PARALLEL FORMS.
One CLAUSE is compared to another CLAUSE.
One VERB is compared to another VERB.
And so on.

Example 1:
The US spends more money on health care than Canada <-- wrong

Interpretation 1: The US spends more money on health care than [the US spends money on] Canada.
Interpretation 2: The US spends more money on health care than Canada [spends money on health care].
While the second interpretation seems to be the intended meaning, both interpretations are reasonable.

The US spends more money on health care than Canada does <-- right

One purpose of do/does/did is to stand in for an ANTECEDENT VERB.
Here, does is standing in for spends.
The addition of does makes the intended comparison crystal clear:
The US spends more money on health care than Canada spends [money on health care].

Example 2:
The critic XY described the performance as more important to popular music concert history than was any other performance<-- wrong

Here, was (verb) lacks a preceding parallel form.
Since was must be compared to a preceding verb, the sentence above is not viable.
The following would be correct:
According to the critic, John's performance WAS more important to popular music concert history than WAS any other performance.

Interpretation 1: The US spends more money on health care than [the US spends money on] Canada.
Interpretation 2: The US spends more money on health care than Canada [spends money on health care].
While the second interpretation seems to be the intended meaning, both interpretations are reasonable.

This is a great explanation Mitch!

Ok, first things first: Thanks Mitch for your response!

So the comparison must be clear, and if you want to add an word/verb it has to be "parallel" to the previous verb. That Rule works for me

But:

Interpretation 1: The US spends more money on health care than [the US spends money on] Canada.
Why/How on earth, should one country spend ON another country??

Do I just assume to much of common knowledge or what?

confused13 wrote: Why/How on earth, should one country spend ON another country??

This phrasing is unlikely to appear on the GMAT.
In a less formal setting, to spend money on country X would mean to devote financial resources to country X.