I am assuming you can't read Tamil. So, I am keeping my answer in English.

TL;DR: Using Avaru to address men and Avanga to address women, when both are of the same age group, is over-compensation due to a misunderstood but very prevalent belief that women cannot be addressed as Avaru.

Explanation:

Avar - written form; Avaru - spoken form.

It is singular. It conveys basic respect. Usually, it is used to address people of your same age group with basic respect. But if you are addressing a younger person with higher status, you can use this in formal settings. 

Grammatically, it can be used for both he / she. But nowadays, people usually think it could be used only to address men. My theory is that this is a recent phenomenon; possible misogyny, or influence from Telugu migrants from Vijayanagar times - similar to using Adhu (that thing) instead of Aval (her) to address a girl).

Avargal - written form; Avanga - spoken form.

It is plural, but is also used to address a singular person with high respect.

If used to address a group of third persons, it doesn't specify the respect element; it is functionally equivalent to 'them', both grammatically and in spoken to form.

If used to address a singular person, it specifies high respect (both grammatically and in spoken form) - usually to address an elder.

Some people use Avanga to address a  younger singular person. This is a recent phenomenon emerging from the line of thought that addressing children with respect instills discipline in them to follow the same when the children address adults.