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What is the differences between Chinese Mandarin and Chinese Cantonese ?

The difference between Mandarin and Cantonese are like difference between English & German or French and Spanish. I mean that they are related languages, but the difference between them is strong. For example, Mandarin has only four tones (mā, má, mǎ, mà), while Cantonese has six. And the whole pronunciation, grammar structure and vocabulary differs too. So usually when a Mandarin-speaker hears Cantonese, he doesn't understand it or he can differ some similar words, because both Mandarin and Cantonese use the Chinese characters, so they have the same vocabulary.
Cantonese sounds more like Thai or Vietnamese. Cantonese is only spoken in Hongkong, Makau, some southern provinces in China (like Guangdong) and among overseas Chinese in the USA, Canada etc. Mandarin is spoken in all over China, Taiwan, Singapore.

Some of the common phrases are written with the same characters in both Cantonese & Mandarin:
歡迎 / 欢迎 (Welcome) M: [huānyíng] C: [fùn-yìhng]
你好吗！(How are you?) M: [nǐ hǎo ma] C: [néih hóu ma]
我愛你 (I love you) M: [wǒ ài nǐ] C: [ngo oi nei]
再見 (Goodbye) M: [zàijiàn] C: [joigin]
生日快樂 (Happy birthday) M: [shēngrì kuàilè] C: [sàangyaht faailohk]
新年快樂! (Happy new year) M: [Xīnnián kuàilè] C: [San1 nin4 faai3 lok6! ]
小心! (Look out) M: [Xiǎoxīn] C: [Síusām]
中華人民共和国万歳！(Long live the People's Republic of China) M: [Zhōnghuá rénmín gònghéguó wànsuì], C: [zungwaa janman gungwogwok maanseoi]

However, there are some differences in vocabulary. F.e. in Cantonese instead of 早上好 [zǎoshang hǎo] they say 早晨 [jóusàhn], which in Mandarin (zǎochen) means just "morning", but doesn't act as a greeting. Other examples:
"Sorry". M: 對不起 [duìbuqǐ] C: 對唔住 [deuim̀hjyuh]
Cantonese 唔 (m) is used instead of Mandarin's 不 [bu], 係 (haih)=是(shi), 乜野 [mātyéh] = 什么 [shénme]. So in Cantonese there are a lot of characters that are rarely used in Mandarin.
"What's your name?". M: 你叫什么名字？[Nǐ jiào shénme míngzì?] C: 你叫乜野名呀？ (néih giu mātyéh méng a)
“I'm not Chinese”。M: 我不是中国人 [Wǒ bù shì zhōngguó rén] C: 我唔係中国人 [ngo m hai zunggwokyan]
"Where do you come from?" M: 你从哪里来？[ni cong na li lai] C: 你係邊度嚟? [Nei5 hai6 bin1 dou6 lai4?]
"I don't speak Cantonese" M: 我不會說廣東話 [Wǒ bù huì shuō Guǎngdōnghuà] C: 我唔識講廣東話. [Ngóh m̀h'sīk góng Gwóngdōngwá]
I've been learning Cantonese for 1 month 我學廣東話一個月嘞。M: [Wǒ xué Guǎngdōnghuà yī gè yuè le] [Ngo5 hok6 Gwong2 dung1 waa6 jat1 go3 jyut6 laak3]
"Where is the toilet?" M: 廁所在哪里？[cèsuǒ zài nǎlǐ] C: 邊度有廁所? [Bīndouh yáuh chi só?] - even the word order is a little bit different here. Mandarin = (toilet)+(in)+(where), Cantonese = (where)+(has/have)+(toilet)

The numbers are pronounced differently as well:
1 M: yi C: yāt
2 M: èr C: yih
3 M: sān C: sāam
4 M: sì C: sei
5 M: wǔ C: ńgh
6 M: liù C: luhk
7 M: qī C: chāt
8 M: bā C: baat
9 M: jiǔ C: gáu
10 M: shí C: sahp