Greenestamps' answer of 480 is correct.  Ikleyn has a little trouble with
English, as it is not her first language.  If '1 appears NEXT to 2', then
and only then, 'numbers 1 and 2 appear next to each other'.

Here is the way I approached it.  I went in the "back door".

Let's say each of the numbers have exactly 1 mate.
1,2 are a mated pair, 3,4 a mated pair, and 5,6 a mated pair.

We will now enumerate all ways in which no two mates are together and then
subtract that result from 720.

We can choose the first number 6 ways.
We can then choose the second number 4 ways, as any of its 4 nonmates.
That's 24 ways to pick the first two numbers. 

In each of those 24 cases, the remaining four consist of a mated pair and a
non-mated pair. 

So there are two cases for picking the third number.

Case 1.  We pick the third number as the mate of the first number in only
one way.

Of the three numbers left, two of them are mates and one is a non-mate to
them.  So the non-mated one must go between them.  There are only 2 ways to
do that, i.e., to place the mated pair on each side of the non-mated one.
That's only 2 ways for case 1. 
 
Case 2.  We pick the third number as a non-mate of the first number (and of
course a nonmate to the second number) in 2 ways.

Of the three numbers left, 2 are nonmates to the third number. So we can
choose the fourth number in two ways.

Now the last two remaining numbers are neither mates themselves nor are they
mates to the fourth number, so they are free to be placed 2 ways, in 5th and
6th positions.

That's (2)(2)(2)=8 ways for case 2.

So that's 2+8=10 ways for any of the 24 ways to place the first two numbers.

So the total number of ways no mates are together is (24)(10) or 240 ways.

The means the answer to the problem is 720-240=480 ways.

Edwin