I think all of the suggestions posted above are valid.
Back in my extract days, I would notice the "twang" in some brews, but not others. When it did appear, I never considered it "cidery"...it came across more along the lines of Dave's description, being kind of caramel/bananna (mostly the former). But other times (even using the same brand of extract) here would be no trace of such 'off' flavors. Also, 35-40 years ago, the available dry yeasts were not very consistent and unless two or even three packets were pitched, underpitching was a problem. And I would agree with EHall that underpitching and stale extract would both contribute to problems (and if fermentation temps were too high, even moreso).
Nowadays, with much better yeast options available and assuming that one's sanitation is all it should be and that good yeast is being pitched in correct quantity, I would point the finger of blame for the dreaded "twang" squarely at the freshness of the extract.
I'm just guessing here, but consider the following:
Extract starts out as wort, and it is concentrated to a comparatively high degree by evaporation. That process alone certainly creates a potential for different flavors forming than what you'd get from a freshly mashed wort. One can look at canned, evaporated milk as an example...theoretically, mixing evaporated milk with an equal volume of water reconstitutes it to whole milk (and the result even meets the legal definition for whole milk). But while it is usable as a substitute and while it is certainly palatable, it does have a much different taste than the everyday carton of milk.