[Zarel]

Here's some that are different:

    方法 (method)
    In Chinese:  fāngfǎ "FONG-fah"
    In Japanese: houhou "HOE-hoe"

    日常 (everyday, ordinary)
    In Chinese:  rìcháng "ZI-tchong"
    In Japanese: nichijou "NEE-chee-joe"

    七月 (July)
    In Chinese:  qīyuè "CHEE-yue"
    In Japanese: shichigatsu "SHEE-chee-gah-tsoo"

(Yes, 'shichigatsu' is seriously the Chinese reading of July; the Japanese reading would be 'nanatsuki'.)

And some that are similar:

    開始 (start)
    In Chinese:  kāishǐ "KAI-tsi"
    In Japanese: kaishi "kai-shee"

    第三 (third)
    In Chinese:  dìsān "dee-SAN"
    In Japanese: daisan "dai-san"

(Note that the pronunciation guide is somewhat approximate since certain sounds don't map well and English vowel pronunciation is a mess.)

I've heard that the Japanese on'yomi (Chinese) readings are generally closer to Old Chinese than modern Chinese is. Also, notice that Chinese is tonal and Japanese isn't.

[jon2512chua]

Wow. So what are the times when one would choose one reading over another?

[Zarel]

The general rule is: if there's a kun'yomi that fits, use the kun'yomi. Otherwise, use the on'yomi.

Most commonly, multiple kanji next to each other are on'yomi and single kanji are kun'yomi.