Your heading referred to "Mandarin vs. Cantonese" but your text was "Putonghua vs. Mandarin". As other people have commented, Putonghua and Mandarin are the same.
Mandarin and Cantonese, on the other hand, are different. Cantonese is so different that there have been various discussions about whether Cantonese is considered a dialect or a language in its own right.
My understanding is that formal written Chinese (grammar and vocabulary) is the same for both Mandarin and Cantonese speakers. However, colloquial Cantonese is totally different.
Mandarin is the official language of China, Taiwan and Singapore. Cantonese is used in Hong Kong and Guangzhou. What you want to study first depends on your purpose of studying Chinese.