When is an erratum necessary? A typo, a spelling error etc., in a published article, is definitely not enough for issuing an erratum.
If a mistake destroys a main result, then an erratum is definitely necessary, and the proof should be rewritten, if it's possible to fix.
What about cases in between?  What if a small lemma, remark is not correctly stated?  What about a mistake in the proof that is actually very easy to fix? What about a wrong number in the calculation? What if ... (you name the case) ...?
 A: Checking MathSciNet in the five-year period from 2010 to 2014, there were 1319 entries with "erratum" or "errata" in the title.  Sampling a few dozen of them, most cases involve a misstatement severe enough to affect the statement of a theorem.  In a few cases, something was inadvertently left out, such as MATLAB code or an author's name. For perspective, during this period, there were 525795 items added to MathSciNet. 
A: The evidence on arXiv does not seem to support the assertion that mathematicians are significantly more reluctant to issue an erratum than, say, physicists:


*

*errata
in mathematics (92 items)

*errata
in physics (160 items)
The factor of two in the number of errata between physics and mathematics correlates well with the total number of arXiv submissions (about three times as many in physics than in mathematics). Incidentally, only a very small percentage of arXiv submissions are errata (the total number math articles is about 150,000).
I do notice that many of the mathematics errata are limited to the arXiv version, and not incorporated in the journal. The idea being, I would think, that issuing an erratum for a relatively minor issue is a service to the reader, who will likely consult the arXiv version and will therefore be alerted to the error. This would obviously be unacceptable for an error that invalidates the entire work, but those happen more rarely.
In this connection I find it interesting to note that mathematicians withdraw more papers from the arXiv than physicists, 128 versus 90 in the last year. I presume this says something about the different consequences of a small mistake in the two disciplines.
A: I can't speak for everyone, but since you ask for examples, I've published "errata" at least a couple of times. Once was a paper (in Inventionnes, quite embarrassing) where the statement of the main theorem was false due to a mistake in the proof! Luckily, a small modification of the proof gave a weaker, but still interesting, true result. This comes under your heading of "If a mistake destroys a main result,..." The other situation, which you didn't mention, was a case where one of the main results of a paper was an immediate consequence of an earlier paper that we had not known about. After our paper was in press, someone pointed this out, so we submitted an "addendum". 
A: It happened to me, too. On my thesis. Published in Annals of Math. I got a note from A. Fröhlich saying that in a seminar in London, they hadn’t been able to verify an involved inequality-computation in a proof. Indeed, my computation was completely wrong. Fortunately, the lemma wasn’t essential to the whole except for producing useful counterexamples. What was most embarrassing was that if I had tried to pull the wool over the reader’s eyes, I couldn’t have chosen a better method of deception. I sent in a one-page correction with a replacement-statement that was rather weaker, but still good for the counterexamples. A few years later, I found a correct proof of the original statement.
