Mike is correct that there aren't "standard" speech patterns for either country. Some of the dialects of British English differ far more from the "standard" RP English than American English does. Talk to a Yorkshireman, a Geordie, a Scot and a Liverpudlian, and you'd be pushed to say that they were all speaking the same language - the differences would be far more than between "standard" English and American English.
But with due deference to linguistic variation, there is some commonality to "British English" (which is probably perceived as RP), as there is in American English.
American English, to an English person, brings up a number of references that colour the perception of the speaker.
The English "R" sound is pronounced with the tongue at the bottom of the mouth, whereas the American speaker tends to roll the tongue back in the mouth - this is characteristic of several English dialects, including the West Country and Norfolk, which are sometimes parodied in England as "country bumpkin" accents.
Vowel sounds are clipped for English speakers - in "stamps", the "a" sound is very short, whereas in some American dialects it's a much longer sound - an extreme example might be "stay-amps".
There is also a greater range of vowel sounds - the English "o" sound in "botanist" is a short sound, pronounced with the mouth in an O-shape. This sound is missing from many American dialects, where it's rendered as an "ah" or "oh". Thus "botanist" becomes "baht-anist" and the Somalian capital "Mogadishu" becomes "mogue-a-dishu" rather than "mog-a-dishu".
The same is true of other vowels - compare the pronunciations of "Iraq" on the news services of the two countries. Irrespective of the language the word is written in, the vowel sounds are different in the speech pattern. Although as Mike rightly points out, you can't easily generalise, you could say that the British vowels are shorter and more clipped.
Delivery is also key - in Britain, children's English is quite sing-song with longer vowels and strong changes in tone and pitch. Adult English in the main has a much flatter delivery, with short vowels and less variation in pitch and tone. American English isn't so flat in delivery, which makes it more interesting to listen to, but can sound childish and sing-song in extremis.
I guess another difference is that there's a more consistent set of pronunciations - American TV coverage is notable to an English speaker for the repeated inconsistencies in the pronunciations of words, typically with words that perhaps aren't in common usage so much in American English.
I recently sat through an American technical presentation about computer networking, which included an extensive section on adding "routes" (pronounced "roots") to "routers" (pronounced row-ters). The incongruity between the two words immediately stood out, and began to detract from what was being said.
To an English speaker, the "row-ter" pronunciation refers to a piece of woodworking equipment, and they'd talk about adding "routes" (roots) to routers (rooters). There's a feeling of contiguity between the base word "route", and the association to "router", which sometimes seems to be lost. Compare the pronunciation of "process", procession" and "processor", which should be consistent.
IMHO, it's completely snobbish to detract from the American dialect, and to let the prejudices of the sound take precedence over what's being said. I think that some of the British, and the British media, overuse the stereotype of the country-bumpkin, childish, sound of some American speakers to detract from what they have to say.
Overall, American English differs a lot less from "standard" English than a lot of people make out, and is certainly less different than some of the British regional accents. However, the cultural and historical differences between the two countries has lead to a quite different interpretation of what's essentially the same language.