Re-Indexing on Search setting, What for?
The possibility to re-index makes sense if your structure has changed or you gather information for the index in another way. For example if you want to show on a node a referenced node in another viewmode (e.g full instead of teaser). But this depends on your general setup and how you configured your content types and viewmodes.

The re-index option deletes the current search index. The "run cron" writes the new search information to the search index everytime the cron runs.