Terseness is short, abrupt, irritated. Succinctness is short, clearly explained. They both mean short, but the tones of the words are quite different. '"Leave me be!", he replied succinctly' doesn't work, for example.
To me, this is exactly the kind of nuance that is lost by training yourself to limit ideas to 140 characters.
Terse also means short, clearly explained. Yes, 'terse' has an extra meaning, but in a technical context, and especially when contrasted with 'verbose', it should be completely clear which meaning I meant.
And for the record: I don't train myself to limit my thoughts, I train myself to use a form for expressing them appropriate to the medium I'm doing it in. Expressing something shorter and in steps is not inferior to long-winded explanations. If you must blame something for a fine nuance I failed to grasp, blame the fact that English is not my first language. I doubt I would have caught it in Dutch either, but at least that could be a valid reason.
But herein lies part of the problem - at first you said that the two meant the same thing in common parlance, then you said the same thing in technical parlance; there's confusion in the audience right there. Anyway, while it was clear what you meant, what you said was not quite correct, which is why I piped up.
Particularly in technical writing, you've got to get your words nailed down correctly. This is why there are things like the RFC defining what 'shall', 'should', and 'may' mean (intended for other RFCs, but it's also used elsewhere)[1], or the development of Simplified Technical English[2]. The latter was developed in aerospace engineering, where minor misunderstandings can literally cost lives.
Subtle differences between words are important, and they allow us to communicate more complex ideas. Twitter may have it's place, but any conversation involving nuance really isn't suited for it. A high-quality conversation includes caveats and careful framing, which can't really be fit into 140 characters. At least, in my opinion.
(That's exactly 140 characters for you :) )