Two-dimensional S-Box Consider the following simple cipher:
$$c_1 = S(m_1 \oplus k_1) \oplus k_2$$
Where $S$ is S-box, $m_1$ - 16-bit plain text, $k_1$ and $k_2$ is two parts of 16-bit key. 

If S-box is standard then this cipher is quite easy to hack by differential analysis. But consider $m_1$ as two-dimensional array 4x4 bits and S-box as combination of two same 4-bit S-boxes. In the first S-box input is 4 rows of plain-text, in the second S-box input is 4 columns of output of first S-box:
$$c_1 = S''(S'(m_1 \oplus k_1)) \oplus k_2$$

Can you help me analyze this algorithm against the following problems:


Are there any advantages and disadvantages of this algorithm to the following:
$$c_1 = S(m_1 \oplus k_1) \oplus k_2$$
where $S$ is a 16-bit S-box.
How resistant is this algorithm to differential analysis.
Is this "two-dimensional s-box" is used somewhere?
I understand the question as you have a single 4-bit S-box, which you first apply rowwise, and then columnwise. 

As already mentioned, this is equivalent to a large S-box $\mathcal{S}$
$$ 
c = \mathcal{S}(m\oplus k_1)\oplus k_2.
$$

This is a well-known Even-Mansour cipher, and it can be broken with complexity $2^{n/2}$, which is $2^8$ for your $n=16$. The idea by Daemen is to try $2^{n/2}$ plaintext pairs with the same difference $\Delta$ both for the entire cipher $E$ and the S-box $\mathcal{S}$.

So the answers to your questions are


No advantages.
Breakable in seconds.
Maybe in some marginal ciphers or hash functions. Note that it is quite inefficient in software.