Asking for anything in Japanese requires you to consider two things:
1. Relationship to the listener
2. Nature of the subject matter
The Japanese concept of 迷惑 (meiwaku) as mentioned above by @MarkHosang is a constant theme in Japanese communication. When the train is late by one minute or neighbors don't separate their trash, this concept of 'causing trouble to others' comes up.
Manners When Taking Leave - 休暇を取るときのマナー
I found this interesting yet brief outline of "[the correct] manner when taking leave." http://www.tisyk.com/cat0201/037/
Basically the page mentions a 休暇届け (きゅうかとどけ) or Leave Request Form. Still used in some old fashioned companies, and perhaps outmoded by email, it is a formal request to your 上司 (じょうし) or boss to "consider across the entirety of the workplace [the effect of your leave] on the progress of work".
The site suggests preparing for your leave by getting your work in order and passing responsibilities to colleagues in advance of your absence, as well as notifying suppliers/vendor companies of your absence.
Basically nothing you wouldn't do in a non-Japanese company, but like most things Japanese, the importance placed on these actions is higher than in most other countries.
The Usually Casual Staff Member
As you might guess from seeing this website, most Japanese people feel like they're burdening their colleagues when taking leave, and so are causing them 迷惑. Even if you don't particularly care, you know that it's costing you brownie points!
In the case of your fellow employee who is usually very casual, might it be fair to extrapolate and say s/he is casual AND less careful with how much 迷惑 s/he causes to other staff members? I don't know.
But if this is the case, then perhaps s/he's being "extra sweet" in asking because s/he is trying to smooth it over. The most まじめ (majime) or serious workers aren't always polite, but it often goes hand in hand.
SIDE NOTE: Although most Japanese companies give 2 weeks of summer holidays per annum, actually taking holidays is at the company's discretion. Friends of mine have had leave requests turned down, even when it meant they didn't get to take the bulk of their leave days. That's reason why it's often nicer to work in 外資系 (gaishikei) or foreign owned companies in Japan: because they usually DO let you take all your leave days.