It seems to me that there are two interconnected problems raised by the formulation of the question. I think it would help to disentangle them:
"meek" v. "humble"
The question of contrasting "meek" and "humble" is bound up with changing English usage. "meek" tends to be somewhat quaint in usage, and certainly not so prevalent in English usage as it once was. Still, its meaning is not very different from "humble", as the Wordnet associations suggest.
My hunch is that this is a bit of a "red herring" here - that is, the important question raised by these texts is not one of "humility" v. "meekness" - a glance at a modern translation like the NASB bears this out. In modern parlance, the "meek" rendering pretty much disappears.
So the answer to this part of the question:
What is the difference between meek and humble?
is that there isn't much if any difference -- in "biblical" parlance -- between these two renderings. The former is a slightly archaic term for which the latter is now more commonly used. It's hard to find "meek" in the fresh modern translations, although it persists in the Authorized Version "tradition" (AV, RSV, ESV, etc.).
How to translate עָנָו = ʿānāw?
OP's question does raise an interesting issue, however, about how עָנָו = ʿānāw, which is the underlying Hebrew word, ought to be translated in any given instance. It is one of a number of terms which can have either a "psychological" or "concrete" nuance. Take these English examples:
(1) He was desolate after her death.
(2) The countryside lay desolate after the invasion.
One can imagine the situation in which a translation would require different words in the target language for what is a common word in the source language. "He" in (1) was not void of vegetation and depopulated; the "countryside" in (2) was not bewildered and incapacitated by grief. (In fact, the Hebrew word שְׁמָמָה šĕmāmâ gives translators precisely this problem!)
So too here with ʿānāw. It is used of either (a) an attitude of pious subjection (so, "humble"); or (b) being bowed down by oppression (so, "afflicted").
The lexica all wrestle with how to make this distinction. Brown-Driver-Briggs can be conveniently given as an example (see top of right column on p. 776). The decision is contextually driven, but there will be contexts in which (as so often for BH.SE regulars) it is a judgment call as to which is in view, and the lexica may well assign a given occurrence to differing sub-entries.
This particular case is further complicated by there being a related and graphically almost identical term, עָנִי = ʿānî, the waw/yod distinction being very fine (virtually identical in the script of many of the Dead Sea Scrolls) which means the "(afflicted) poor", where it is the poverty that seems to come to the fore.
In sum, the very short answer to OP's question regarding the Hebrew word:
Why did the translators choose two different words?
is context! It is one of those knotty problems for translators.