sick及ill有何分別？為何病假不是ill leave？

SOME and ANY
 
Some is generally used in affirmative sentence; any is used in questions and negatives.
I want some money.
Have you got any money?
Sorry, I haven't got any money.
 
Some is used in questions when we expect an affirmative answer.
Could I have some rice, please?
Would you like some more tea?

Any is used in affirmative sentences that really have a negative meaning, with words like never, without, hardly.
You never give us any help.
There is hardly any Chinese tea left.


ILL and SICK

(1) Sick can be used to talk about bring up food from the stomach, or wanting to do this. Be sick sometimes means “vomit”.
I felt sick once the roller coaster started to move.

(2) Sick or ill, the words are to an extent interchangeable. One can say he is sick or he is ill. Ill suggests something serious like a stroke whereas sick suggests a sore throat. Overall, ill is more serious, sick is more trivial.
But non-humans tend only to be sick e.g. a sick cow, a sick joke, a sick economy.

(3) Sick of = tired of; fed up with
I am sick of your complaining. (Ill would not work here.)

Be careful with the following usage:
He can ill afford to buy a new car now. (He is in a poor financial position now. )
The kitchen now becomes totally ill equipped. (Ill equipped means badly equipped).
seriously ill (more natural than seriously sick.)
sick leave (not ill leave).
called in sick (not called in ill)
Seasick, carsick
Ill-treated, ill-equipped