卵 vs 玉子
生花 vs 生け花

Why?
The same meaning but different ways to write it down. 
Is this just a question of personal writing style?
Maybe depending on region, age, ...?

[Zgarbas]

卵 vs 玉子
卵 is a raw egg.
玉子 is a cooked egg.


生花 vs 生け花
both are correct because okurigana can be quite random, but they're not exactly that different; it probably started out with 生け花 as the correct form and people got lazy and started skipping the middle part since the reading is still obvious.

Some googling of ”なまたまご” and ”ゆでたまご” with different combinations of kana and kanji gives the impression that usage is all over the place (like I expected). What I can say is that 卵 (to me) feels more formal/scientific and that 玉子 feels more informal/culinary. Not to say that there's anything wrong writing it in hiragana either. If anything 生花 feels more old fashioned, since consistent okurigana and the like is pretty modern, but I could be wrong.

卵 is any kind of egg, 玉子 is specifically a chicken egg. Whether it has been cooked or not.
That's according both to EDICT and Shogakukan (via goo.ne.jp).

There's no -general- answer to the question, however. Sometimes the characters are used for slightly different meanings, but sometimes it's just arbitrary. Most of the time there at least -was- a usage difference even if it got lost with time, but there are also times when it's simply that different scribes in different parts of Japan chose different characters to write the word with. Most of those differences got normalized and only one choice survived, but sometimes two (or more!) ways to write a word continue to this day.

You can generally find out the answer for any specific word - if it's not in the dictionary, someone has probably already asked on chiebukuro. If they haven't then you can ask yourself.