I've more sentences about it because I (BrE) go about/(AmE) go around discussing it in my talk: "How America Saved the English Language". It's one of a long list of differences for which the folklore is faulty, with people like comedian David Mitchell (below) assuming and repeating that Americans don't pronounce the 'h' in herb because we think we (or the word) are French. (The implication here is that the British are not under the illusion that they are French. Except of course that they eat aubergine rather than eggplant and increasingly use -ise instead of -ize and spell centre with the letters in a very French order. And so on. And so forth.)
Mitchell went to Cambridge University, apparently (according to his Wikipedia bio) because he was rejected by Oxford. I can only assume this has caused him some sort of allergy to the Oxford English Dictionary and that this caused him not to research the claims he made here about herb as well as tidbit/titbit. Had he just looked it up, he would have found the following information.
From the Middle Ages, the word in English was generally spelled (or spelt, if you prefer) erbe, from the Old French erbe--but sometimes it was spelled with an h, after the Latin herba. From the late 15th century the h was regularly included in the spelling in English, but it continued not to be pronounced for nearly 400 years. This was not a problem for English, of course. We often don't pronounce written h, for example in hour and honest and heir, and our ancestors didn't pronounce it in humo(u)r, hospital, or hotel. Change and confusion about these things leads to the oddity of some people insisting that some (but not other) words that start with a pronounced h should nevertheless be preceded by an, not a, as if the h weren't pronounced. (AmE) To each his/her own/(BrE) each to his/her own...
The h in herb finally started being pronounced in the 19th century in Britain. By this time, the US was independent and American English was following a separate path from its British cousin. Why did the English start pronouncing it then? Because that's when h-dropping was becoming a real marker of social class in England. If you wanted to be seen as literate (or at least not Cockney) you had to make sure that people knew you lived in a house, not an 'ouse. This 1855 cartoon from Punch (reproduced as a postcard for the British Library's Evolving English exhibition) illustrates:
The result seems to have been more self-consciousness about pronouncing h where it was in the spelling, and some h's got louder where they had not previously been heard. Why did this happen to herb and hotel but not honest or heir? I don't know.
So, pronouncing herb without the h is the Queen's English, if we're talking Elizabeth I, rather than Elizabeth II.
And in case you were wondering: Americans pronounce the h in the name Herb, which has a different history from the plant herb.