1. Results close to as good if not better than Google.
2. No tracking.
3. Bang searches. With the "too clever by half" search dialogs many sites are implementing (Wikipedia, Internet Archive, Google Ngram Viewer) it's often superior to the native on-site search as I don't have to set focus 2-4 times to actually type in the bloody search dialog.
4. No tracking.
5. Did I mention no tracking?
6. Proxy bang searches, e.g., StartPage.
7. Search customisations -- light page (good for console and commandline search), night mode, and others.
8. Unmangled URLs. I can copy/paste URLs straight from DDG. No-can-do with Google.