What is the meaning of the weights in a CES-production function? In different paper I offen encounter two types of CES production functions

$y=\left[\left(1-\omega \right) x_{1}^{\frac{\sigma -1}{\sigma}}+ \omega x_{2}^{\frac{\sigma -1}{\sigma}} \right]^{\frac{\sigma}{\sigma -1}}$

or

$y=\left[\left(1-\omega \right)^{\frac{1}{\sigma}} x_{1}^{\frac{\sigma -1}{\sigma}}+ \omega^{\frac{1}{\sigma}} x_{2}^{\frac{\sigma -1}{\sigma}} \right]^{\frac{\sigma}{\sigma -1}}$

the only difference being the exponents of the weights.

What is the difference in meaning and interpretation of the two weights (on an intuitive and a theoretical level). Does it matter which version to use in a paper?
The weights are usually called "distribution parameters" (top of page 12 of linked document), as they correspond, in the case of a Cobb-Douglas, to each factors share in total output. For example, if the two factors were labour and capital respectively, $(1-\omega)$ would be the labour share, and $\omega$ would be the capital share. Because in the case of a Cobb-Douglas ($\sigma=1$) both specifications are identical, the weights refer to the distribution parameters in both specifications.

To see this, take the second equation. First, define the share of $x_1$ in total output as

$$ \lambda_1 \equiv \frac{w_1x_1}{py} $$

where $\frac{w_1}{p}$ is the real return to factor $x_1$ (e.g. real wage). 

Under competitive conditions, the firm(s) employ $x_1$ until its marginal product equal $\frac{w_1}{p}$. This marginal product is:

$$ \frac{\partial y}{\partial x_1} = (1-\omega)^{\frac{1}{\sigma}}\left(\frac{y}{x_1}\right)^{\frac{1}{\sigma}} = \frac{w_1}{p} $$

Combining the later equality with the definition of the factor share, you get that

$$ \lambda_1 = (1-\omega)^{\frac{1}{\sigma}} \left(\frac{y}{x_1}\right)^{\frac{1-\sigma}{\sigma}}  $$

In the case of a Cobb-Douglas ($\sigma=1$):

$$ \lambda_1 = (1-\omega) $$

Equivalently, 

$$ \lambda_2 = \omega $$

which means that $\lambda_1 + \lambda_2 = 1$, in line with the assumption of constant returns to scale, in both specifications.

Regarding the difference between the two, according to my calculations, the second specification does not encompass the Leontieff case ($\sigma = 0$). This is because, whereas in the first specification you can solve the limit of $(1-\omega)x_i^{\frac{\sigma-1}{\sigma}}$ when $\sigma \rightarrow 0$ (using the minimum of $x_i$, see equation (23) here), you cannot do the same for $(1-\omega)^{\frac{1}{\sigma}}x_i^{\frac{\sigma-1}{\sigma}}$, because the first component goes to zero (provided the standard assumption $0&lt;\omega&lt;1$).

Thus, if you want to be as general as possible, I would recommend the first specification. Additionally, whereas the first specification is imo "very common", the second is not, which would require further justification as to why you choose it. If there is no extra benefit of it, then, by Occam's Razor, the first one is definitely preferred.