The difference is a very simple one. It has nothing to do with grammar - it's purely a question of writing two words as if as one.

The word 'another' is just 'an + other'. It's as simple as that.

The book/the other book

The books/ the other books

That book/ that other book

Those books/ those other books

My book/my other book

A book/ a other book / an other book / another book

Easy, isn't it? There's no big mystery - it's just an+other

A while ago, somebody here started a Discussion with a question along the lines of 'If you could change one thing about your language, what would it be?'. My vote was to change the spelling of 'another' back to 'an other' - two separate words. If it were written as two words, it would save everyone -students and teachers alike - so much time and trouble.

You mean another and other. 

"Another" is addition to the current amount. Whereas "other" means different than the current object. 

For example: " I'm going to buy another pen."  It refers that you've one. " You should pick up the other one" it refers that you should pickup the one other than that you are picking. 

We use "another" with singular nouns. An = one + other.

We use "other" with plural nouns.

For example:

Another continent I want visit is South America.

Other countries I want visit are Brazil, Peru and Chile.