[Update, 14 June 2006: As is often the case, Americans have the older form of the word--the British used to say 'erb too. It just happened to be mentioned in the Guardian's Weekend magazine this week. See Michael Quinion's World Wide Words for more...]
[Update, 3 September 2014: I've now done a proper post on herb.]
A common response to an American pronunciation of herb is: "Are you a Cockney, then?" Dropping aitches is a definite marker of lower social class--and these days it's fairly rare. In fact, aitches get inserted sometimes in the name of the letter, i.e. haitch. This is heard in the semi-humorous admonision to not 'drop your haitches' (and thus sound 'common'), but is heard unironically in many people's everyday speech, although it is not considered to be 'standard' usage. The story is that it's the Irish pronunciation, and I've read in various places that haitch marks Catholics in Northern Ireland and the Catholic-educated in Australia. I've noticed no such associations here, and neither have friends of mine, though one did suggest that it might be a marker of region rather than religion here. Indeed, my haitch-saying friend is from Liverpool, whose dialect (Scouse) is influenced by Irish immigrants.
As long as I'm talking about herbs...there aren't many that differ in name between the US and UK. Americans call the green part of the coriander plant cilantro, while the British call it coriander. Americans use coriander to refer to the spice made by drying and grinding the plant's fruit. Presumably the difference exists because Americans were introduced to the herb in Mexican cooking, whereas the British know it from South/Southeast Asian cooking. Once, reading a British recipe in Texas, I got confused. I knew that British coriander wasn't meant to refer to the powder in my coriander jar, but could only remember that the American translation also started with C. So I put a whole lot of cumin into my chicken soup. I ate about three bites before I decided that there was nothing to do but toss it out.
Oregano differs in pronunciation, with Americans saying oREGano and the British saying oreGANo. In South Africa (where I first started picking up 'commonwealth English'), they use oreGANum.
As for other herbs and spices, I have been asked "Why do Americans put cinnamon on EVERYTHING?" I can only answer (ignoring the hyperbole): "Because it's tasty."