...is it same as I increase ISO when capturing the photo? If they won't be the same, what's the difference...

The end result is similar, but how you get there and the side effects are different.

Increasing the ISO setting on the camera results in the addition of gain (amplification) in the path between the sensor and analog-to-digital converter, which is what turns the light levels into numbers. This is done with analog circuitry which amplifies the signals that make up the image to make them brighter. The trade-off is that this process also amplifies any noise in the signal, and that ends up in your image even if you don't make any adjustments in post.

Making exposure adjustments in post requires a bit more explanation:

For the sake of simplicity, say your camera produces grayscale images containing pixels whose light levels range from 0 (black) to 100 (white). A properly-exposed image will have its light levels cover this entire range, giving you 101 distinct shades of gray. (See this tutorial on histograms for further explanation of what this means for your photos.) An underexposed image will cover a smaller range of those values, say 0 to 50, because the amount of light falling on the sensor isn't enough to drive any of the pixels all the way up to 100. Even though the camera can record 100 different shades, this particular example records only 50. Fewer shades means fewer details in lower-contrast areas of the image. The more you underexpose the image, the worse this problem becomes. (Tangentially related is posterization, which is when this is done on purpose.)

Making a positive exposure adjustment with your processing software multiplies each pixel value by some value. In the example above where the brightest pixel value was 50, multiplying by 2 would bring that to 100. This results in an image that appears better-exposed because it covers the full black-to-white range. Because the software can't recreate information the camera didn't record, you're still working with the same 50 shades, and that lack of information becomes easier to spot in the adjusted image.

...or which way is better?

Neither. The ideal for any image is to expose it at the lowest ISO possible (to minimize noise) and long enough to make sure the image's histogram covers the entire light-to-dark range (to maximize detail). Anything else is a compromise that requires weighing what you're shooting with the kind of results you can live with. All other things being equal, you can get more detail by settling for a higher noise level or a lower noise level by settling for less detail.

If I had to pick one over the other, I'd opt for higher ISO, but that's because the body I shoot with does well in the dark and the camera-profiled noise reduction in my processing software does a decent job. Your camera and software may behave differently, and the only way to know which option is right for you is to experiment.