Very little. It is like the differences--probably less than the differences--between the Spanish of Peru, Argentina (vos verb form), and Spain (ceceo pronunciation, vosotros verb form). 

There are small differences in vocabulary. US/UK: elevator/lift, gas/petrol, diaper/nappy. It's like the differences carro/coche, computadora/ordenadora, and so forth.

There are small differences in pronunciation ("accent.") For example, US speech is "flat," UK speech is more "musical." However, the differences between "General American" English and standard British English are smaller than regional differences within either country.

The spelling differences can be traced to one man, Noah Webster. In the 1800s, the United States had separated from England, and we were a literate nation with newspapers, printing presses, and publishers. Webster decided that the United States should have its own dictionary and produced An American Dictionary of the English Language. For whatever reason, he decided to simplify and rationalize a few spellings here and there. In my opinion, it was stupid, because he didn't simplify it enough to make it better, he just changed it enough to make it different. 

In my opinion, the existence of two separate dictionaries--Merriam-Webster in the US, Oxford in the UK, with different spellings--has resulted in an exaggerated idea that there are two "different" kinds of English. 

The Dictionary of American Regional English, which records the difference in English spoken in different parts of the US, is six volumes long and took more than thirty years to complete. Yet nobody asks "should I learn Californian English or Michigander English?" 

For the purposes of learning they are quite similar. Native speakers usually understand both but have slightly different word choice and pronunciation habits depending on the area of the world they come from.

My thoughts are that if you're going to have a strong connection with one particular country then learn the English from that area first.

However, if you would like to understand books and media and talk with a wide range of people then you're going to need to understand both - most native speakers understand more-or-less all forms of standard English and definitely the British and American standards. It's only really the more regional accents/dialects that differ from the standard quite a bit - mostly these are from areas that don't produce much media (that makes it into the outside world) so most native speakers haven't been exposed to those accents/dialects either.

So for listening you're best to learn from a wide variety of voices.

When speaking, you'll just need to pick one for yourself. It's not going to matter what you pick until you get very good at English and you're trying to sound like a native speaker (you're going to sound foreign up till then so your main goal should be clarity).