Both seem to make image sharper. Are those based on the same approach? If they're not, what's the difference?

Just curious.

Yeah, in general sharpening filters increase the contrast of the small (local) details in an image. The type of algorithm used for finding those local detail is often a convolution / kernel filter: https://en.wikipedia.org/wiki/Kernel_(image_processing)
In short, it compares every pixel in the image with its neighboring pixels and then calculates a new pixel value from it, and from that it can enhance (sharpen), extract (edge-detect) or reduce (blur) the local differences in the image. To find larger details it has to consider a wider radius of surrounding pixels (larger kernel size), which also takes longer to calculate.

A high-pass filter essentially gives you the separated details in a layer, so using a blending mode like Overlay, Linear Light or Grain Merge will act as sharpening, while for example Grain Extract (the opposite to Grain Merge) will result in blurring. The reverse is also how you create your own high-pass filter: Add a Filter Layer above your image, choose Gaussian Blur and play with the radius, then set its layer blending mode to Grain Extract.