By Contrast/details I understand that you mean Contrast by Detail Levels.
Following the explanation given by @afre, I wish to further differentiate between them:
- capture sharpening: it’s always a must use tool
- if you don’t resize your image upon final exporting, then this tool is about all you will need to sharpen your image (always depending on the camera/lens combination inherent sharpness, and on your skills while shooting). Perhaps a bit of extra standard sharpening will be needed, but not much
- if you resize the output image, then you have to look at this tool as a helper for other tools, so they work much better, but most probably you will always have to post-resize sharpen the output image
- contrast by detail levels (CbDL) is a tool that uses wavelets (D2 wavelets) and only allows you to tweak the finest details of the image. Sometimes this is enough, others don’t.
- contrast module within wavelet levels tool: this tool is more comprehensive than CbDL, as it allows to choose between D4, D6, D10 and D14 wavelets, you can tweak up to 10 levels of detail, plus a residual image (what remains after extracting all the details), and has many more modules to work with each level and combined with the contrast module
As a general rule, you may wish to always use capture sharpening (fine tuned for each image) and wavelet levels contrast (alone or in combination with some other modules present there).
The problem is that currently the documentation for wavelet levels is not up to date in English, so perhaps you may get decent enough results with CbDL until the documentation is finally updated.
On the other hand, you can sometimes combine capture sharpening with other tools (Tone Mapping, Local Contrast, …) and get a perceived sharpness increase that feels enough for you.