Why is 两 used for quantities of two rather than 二?
The historical reason behind it is kind of surprising though. From a Chinese Characters Roots book:




  The earlier form [of the character] looked like a yoke and a pair of saddles of a two-horse carriage. The initial meaning was two or double. It was also a unit to count vehicles and written as 辆 later.


That is why we have:


  一辆车  ―  yī liàng chē  ―  a car (from Wiktionary)


Where you notice in 辆 that there is the 车 character, meaning car, at the left side of the 两 character.

I assume its usage became a practical one, since as in many languages (including English), you can be more specific when you are referring to a pair of objects by using 两, instead of two objects using 二.

Here's a particular example: 二炮 (Èr pào) refers to the Second Artillery and 两炮 (liǎng pào) means two guns. So, would you request two guns from your commander or a full regiment?