Buksy’s answer is correct but in my opinion it does not answer the original question, namely: “What can service workers do that web workers cannot? Or vice versa?”
There are fundamental differences in their lifecycle and the number of instances per origin you can have. In short:
| Web Workers | Service Workers |
|--------------|--------------|------------------|
| Instances | Many per tab | One for all tabs |
| Lifespan | Same as tab | Independent |
| Intended use | Parallelism | Offline support |
Buksy’s answer is basically the last row of the table. Credit: I took this table from Demystifying Web Workers and Service Workers by Nolan Lawson, starting from slide 35.
In particular, here is how you spawn and terminate web workers:
whereas service workers have their own lifecycle, which is admittedly their “most complicated part”:
So lifecyle is one fundamental difference between the two (a consequence of their intended use).
There used to be a huge difference in browser support: Service workers were not available at all in Safari for iOS till 11.3 (2018 Mar 29), see Can I use service workers? In contrast, web workers had a much better browser support already in 2012: Can I use web workers?
If you have to support IE11, you can only use web workers: IE11 does not have service workers, and apparently the end of support for IE11 is October 14, 2025.
There are subtle differences in their API support across browsers, see HTML5 Worker Test (also by Nolan Lawson). In a particular browser, one kind of worker might support a certain API call whereas the other does not. Visit that page and test your own browser!