How do "augment" and "increase" differ?

Definition of augment by Dictionary.com:

to make larger; enlarge in size, number, strength, or extent; increase

Definition of increase by Dictionary.com:

to make greater, as in number, size, strength, or quality; augment; add to

I cannot find the difference because the definition of the 2 words are the same.

For example, "her secretarial work helped to augment her husband's income"

Wouldn't it mean the same if I replace it with increase?

From Google's definition:

aug·ment verb ôɡˈment/ 1. make (something) greater by adding to it; increase. "he augmented his summer income by painting houses"

When you use augment, you mean that you are adding to something by adding in something else; the word is generally used with a prepositional phrase starting with by or with. Increase doesn't have that sense.

Now, to your example. If the secretary augmented her husband's salary, that means her salary is added to his, for a greater combined income for the couple. If the secretary increased her husband's salary, that means that she gave her husband a raise (i.e. the extra money would be his salary as well as the original amount) and it would imply that the lady was her husband's boss.

Augmenting has a positive connotation. It sort of implies that we are adding on to something with a desirable effect, or making something more complete.

On the other hand, increasing has a neutral connotation and objectively describes the quantitative change of something.

No, it would not mean the same. The verb augment mainly means:

Make (something) greater by adding to it

To increase her husband's income means the amount of his salary increases. Her income will only add X amount to his salary making them (his and her income) total household income.

His boss (CEO) might be able to increase (raise) his salary for his good performance, but his wife can't.

[Oxford Online Dictionary]