:: COMPARE ::
The short answer
Luck requires nothing of you. Serendipity, as coined, requires two things at once: that you were not looking for it, and that you were sharp enough to see it. Drop the sagacity and you have merely renamed luck.
Side by side
| Aspect | Serendipity | Luck |
|---|---|---|
| Requires alertness | Yes | No |
| Requires not-seeking | Yes | No |
| Can be cultivated | Partly | No |
| Walpole would allow it | Yes | No |
What each one means
Finding something valuable you were not looking for. In everyday use that is simply a fortunate accident; in Walpole's stricter original sense it also takes the sagacity to notice what you have stumbled on.
Full definition of Serendipity →The contrast above sets out how Luck differs from Serendipity on the points that actually matter.