Why are the 'color-neutral' gluons confined? What makes the two 'color-neutral' gluons
$(r\bar r−b\bar b)/\sqrt2$ and
$(r\bar r+b\bar b −2g\bar g )/\sqrt6$ different from the pure $r\bar r +b\bar b +g\bar g $ ?
Why don't they result in long range (photon-like) interactions?
 A: One could state your problem as " why are there only eight gluons accepted as physically existing  from the SU3 symmetry . This link gives the answer simply, and it says that if the ninth all color neutral gluon existed, then as you guess, it would act similarly to the photon with interactions between hadrons that have not been observed.
So the answer is: so that the SU(3) color model should fit the data , i.e. strong interactions between hadrons are confined because we have not observed colorless gluons in the interactions between hadrons.
In the matrix analysis the link goes on to show this :

But, no matter how we take complex linear combinations of trace-zero hermitian matrices, we cannot get

1 0 0
0 0 0
0 0 0

since this has trace 1.  (I.e., the sum of the diagonal entries is 1: that's what the trace means, the sum of the diagonal entries.) So we cannot really get red anti-red.  The closest we can get are things like

1  0  0
0 -1  0
0  0  0

or

1  0  0
0  0  0
0  0 -1

or

0  0  0
0  1  0
0  0 -1

But note, these three are not linearly independent: any one of them is a linear combination of the other two.  So we can get stuff like
(red anti-red) − (blue anti-blue)
and so on, but not 3 linearly independent things of this sort, only 2: one less than you might expect.

It is thus fortunate that SU(3) color can be used to described what has been observed with hadrons, i.e. no neutral photon like  gluons
A: There is no fundamental difference between the gluons $(r\bar{r}-b\bar{b})/\sqrt{2}$ and $(r\bar{b} + b\bar{r})/\sqrt{2}$. The first one is represented by the matrix
$$
Z = \frac{1}{\sqrt{2}}\left(\begin{array}{rrr} 1&0&0 \\ 0& -1 & 0 \\ 0 & 0 & 0\end{array}\right)$$
and the second by the matrix
$$
X = \frac{1}{\sqrt{2}}\left(\begin{array}{rrr} 0&1&0 \\ 1&0&0 \\ 0 & 0 & 0\end{array}\right).$$
However, these two matrices are related by the change of basis
$$
H = \left(\begin{array}{rrr}1/\sqrt{2}&1/\sqrt{2}&0 \\ 1/\sqrt{2}& -1/\sqrt{2} & 0 \\ 0 & 0 & 1\end{array}\right). 
$$
It is easy to check this by multiplying matrices and seeing that $HZH^\dagger = X$.
Thus, if you call $(r\bar{r}-b\bar{b})/\sqrt{2}$ "color-neutral" and $(r\bar{b} + b\bar{r})/\sqrt{2}$ "non-color-neutral", it is clear that "color-neutral" is not a property that is invariant under change of basis, and thus is not a meaningful property in quantum chromodynamics.
Actually, neither of these gluons is color-neutral.
