It's not about meaning - it's about structure and grammatical accuracy.

Sure, you can say either '<em>I want to visit another country'</em> or '<em>I want to visit other country'</em>.  We would understand what you meant whichever one you said.  So, in that respect, they have the same 'meaning'. But the difference is that '<em>I want to visit another country'</em> is correct English, and <em>'I want to visit other country'</em> is poor English. It's like 'pidgin English' - people understand it, it would get your meaning over, but it's grammatically wrong, and sounds bad.

Remember - the word 'another' is simply the article 'an' and the adjective 'other', written as one word. That's all it is.

An + other = another

So, would you say <em>'I want to visit interesting country'</em> or '<em>I want to visit an interesting country'</em>?

The second one, of course, is right. 'Country' is a singular noun, and in this sentence it needs to have an article (a or an) before it.

a country

an interesting country

an other country  --> another country

So when do we use 'other'?  Quite simply, we use it in situations where we don't use the article 'a' or 'an':

Plurals:  other countries, other people, other languages

Uncountable nouns:  other advice, other information

When we have another determiner: the other country, that other country, my other country

I hope that makes sense.

The difference is very simple:

another = an (1) + other

So, you can't use "another" for plurals, and if you use "other" for single things, you need something extra, eg. some other, my other, one other.

They do have similar meanings, but I would say there is a slight difference in using them. This may help you: http://www.grammar.cl/english/another-other-others.htm