In our daily spoken language, they are almost identical. No one cares which word you use. The difference is intonation, and the intonation brings emphasis.
- 儘 (jǐn) 量: lighter intonation of "jin", is asking/answering politely. It implies "try one's best efforts if possible".
- 盡 (jìn) 量: heavier intonation of "jin", is asking/answering intensely/strongly. It implies "try to get the best results if possible".
Here are the metaphors of 3 word 儘/盡/量 in the phrases:
- 量: degree/level/extent; usually measurable or comparable.
- 儘 (jǐn): to the best/most (and try to reach the limit if possible)
- 盡 (jìn): to the limit (and try to push or break the limit as it can)
Your best may not be your limit.
Similarly, there are 儘管, 盡管, and 僅管:
- 儘 (jǐn) 管: to feel free to.
- 盡 (jìn) 管: informal written of 儘管. Using this phrase is just like 盡量: to emphasize this phrase. It implies to break the constraint.
- 僅 (jǐn) 管: nevertheless
Here in those phrases, 管 means "the constraint".
In many cases, 儘量/盡量/儘管/盡管 could be interchangeable in the daily spoken language (but not always interchangeable), and we really don't care about the differences. It's just your speaking style.
For example:
- 你儘量吃: You could/may eat as possible as you can
- 你儘管吃: Please feel free to eat
BTW, the phrase 僅管 and 儘管 sound the same (jǐnguǎn), but they have different meaning. They are not interchangeable in written, but many people use them incorrectly.