The word 'abbreviations' and 'acronyms' are often used interchangeably.
Abbreviation is a version of a word or phrase that has been reduced in length used to represent the whole word.
Example:
For North Americans, math is the preferred abbreviation for the word 'mathematics' .
For the UK, maths is the preferred abbreviation for word 'mathematics' .
Acronyms is a type abbreviation that are pronounced as words. We do not say each letter at a time.
Examples:
Teaching English to Speak of Other Languages - TESOL
Acquired Immune Deficiency Syndrome - AIDS
radio detecting and ranging - Radar
self-contained underwater breathing apparatus - Scuba
You live only once - YOLO
Initialisms, uses the first letter of every word of the phrase, is a type of abbreviation that are pronounced one letter at a time.
Examples:
Federal Bureau of Investigation - FBI (Each letter pronounced separately)
By The way - BTW (Each letter pronounced)
Sometimes Initialisms are also acronyms.
Example:
As soon as possible - ASAP (can be pronounced as one word or one letter at a time).
I hope this clarifies the differences.