What are the differences between these two Gabor filter functions I need to enhance the visibility of veins on dorsal hand vein images in my project. I use two different even-symmetric Gabor filters bank improve vein visibility.

First bank consists of these gabor functions:
$$G^\mathit{e}_\mathit{mk}(x,y)=\dfrac{\gamma}{2\pi\sigma^2}\exp\Bigg\{-\frac{1}{2}\left(\dfrac{x_\mathit{\theta}+\gamma^2y_\mathit{\theta}^2}{\sigma^2}\right)\Bigg\}\times \left(\cos(2\pi f_\mathit{0}x_\mathit{\theta})-\exp(-\dfrac{\upsilon^2}{2})\right)$$

Second bank consists of these ones:

$$G^\mathit{e}_\mathit{mk}(x,y)=\exp\Bigg\{-\frac{1}{2}\left(\dfrac{x_\mathit{\theta}+\gamma^2y_\mathit{\theta}^2}{\sigma^2}\right)\Bigg\}\times \cos(2\pi f_\mathit{0}x_\mathit{\theta})$$

where $m$ is the scale index, $k$ is the orientation index, $f_\theta$ is the filter center frequency, $\sigma$ is the standard deviation (often called scale), $\gamma$ is the aspect ratio of the elliptical Gaussian envelope, $\upsilon$ is the factor determining DC response, $x_\theta=(x\cos\theta+y\sin\theta)$ and $y_\theta=(-x\sin\theta+y\cos\theta)$ are rotated versions of the $x$ and $y$ coordinates.

I have coded these filters in MATLAB, I do not have any problem on coding. But I cannot understand the underlying difference between these two gabor functions.
Depending on the location of the peak and the scale of the two axes of the Gaussian envelope the filter may have a large DC response. A popular approach to get a zero DC response is to subtract the output of a low-pass Gaussian filter, which is what the first of these two does. In the case of images, if the DC response is not removed, the filter will respond to the absolute intensity of the image.

This tutorial gives a bit more detail.