This is an impossible question to answer as there are many very different UK accents. A person who understood no English at all might listen to a London accent, a Liverpool accent and a Glasgow accent, and presume that these three people were speaking different languages. They're not - it's exactly the same language - but the pronunciation is very different indeed.
US accents don't vary quite as much, but there are still a number of variations across the USA. Compare an Alabama farmer to a NYC taxi-driver, and you're sure to hear a big difference.
That said, we can make a few generalisations. One is the letter 'r'. In North American English the letter 'r' is always pronounced. In standard British English the 'r' is not pronounced if it comes after a vowel. For example, for many British people the word 'car' is pronounced with an open 'ah' sound, as if it were written 'cah'. But please don't assume that everybody in the UK says 'car' like this. In Scotland, Northern Ireland, and parts of the west of England, the 'r' sound is very much in evidence.
As a general rule, Americans are more likely to pronounce a word such as 'butter' or 'water' with a 'flap' sound instead of the 't'. This might appear silent, or sound like a soft 'd'. But bear in mind that not all Americans do this, and not all British people pronounce the 't' in these words. There are many regional accents, such as the typical London accent, which also misses out the 't' in 'water' and 'butter', replacing it with a glottal stop.
The best thing for you to do is listen to as many examples as possible, and you'll soon get a feel for the general differences in sound.