What is the difference between "to augment" and "to increase"?
"The mixed burgers had higher levels of certain sulphur-containing compounds that are believed to augment the meat "notes" in a burger's aroma.

"The recent speech of the president augmented tensions in the Near East"

"The pressure augmented" "Continuing rains augmented the floodwaters."

http://www.thefreedictionary.com/augment
1. to make larger; enlarge in size, number, strength, or extent; increase.
2. Music. a. to raise (the upper note of an interval or chord) by a half step.
b. to double the note values of (a theme).
3. Gram. to add an augment to.
4. to become larger.
5. a prefixed vowel or a lengthening of the initial vowel that characterizes certain forms in the nonpresent inflection of verbs in Greek, Sanskrit, Armenian, and Phrygian.

Question

What is the difference between "to augment" and "to increase"?
"The mixed burgers had higher levels of certain sulphur-containing compounds that are believed to augment the meat "notes" in a burger's aroma.

"The recent speech of the president augmented tensions in the Near East"

"The pressure augmented"         "Continuing rains augmented the floodwaters."
 
 http://www.thefreedictionary.com/augment
 1.  to make larger; enlarge in size, number, strength, or extent; increase. 
 2.  Music. a.  to raise (the upper note of an interval or chord) by a half step. 
                  b.  to double the note values of (a theme). 
3.  Gram. to add an augment to. 
4.  to become larger.   
 5.  a prefixed vowel or a lengthening of the initial vowel that characterizes certain forms in the nonpresent inflection of verbs in Greek, Sanskrit, Armenian, and Phrygian.

Answer

The first example "augment the meat notes" is good. However, I would use "increase" when talking about the tensions and the floodwaters.
Generally, "increase" is much more common than "augment." "Augment" usually refers to increasing something that is desirable, whereas "increase" can be used in all contexts. The dictionary definition you quote gives you some technical uses of the term in music and linguistics, where we cannot use "increase."
In your first example, you could also use the word "enhance."

Answer

There is no difference.  The two words, "to augment" and "to increase" mean the same thing.