I've had various requests to cover various telephone-related vocabulary. Most of it is simple enough that I can do that thing that I think of as not-really-writing-a-legitimate-blog-post--that is, writing a big list of equivalent words. Some aspects might prove harder, though. Take, for instance, this email from someone I know:
I had a proposal from a US Co. today. For a British English speaker it was virtually incomprehensible unless you knew (which I didn't [BrE] twig [='understand'] until I'd read it for the 6th time) that a 'deck' was a mobile phone and a 'carrier' was what we call a service provider. The most unintelligible phrases included the statement that 'Carrier WAP-deck retail space largely dictates sales' and a sentence about 'On-deck carrier competitions'.In this case, I think we're looking at more than a BrE/AmE difference. Deck, as far as I can tell, is industry jargon for a phone as a platform for a game.  Searching the web for "receive calls on your deck" gets zero matches (versus 232K matches for "receive calls on your cell").  It's not impossible to find this on UK websites (e.g. this one). So, I'm not convinced that that deck belongs here.

But there are enough others that do belong here. So, here's the list. No, wait! Here's the preface to the list:
 Items in [square brackets] are found in both dialects, with no indication in the OED that it is original to the dialect whose column it's in. Nevertheless, its counterpart in the other column is specific to that dialect.Items marked * are found in the other dialect now too, though they are not original to it. I haven't included really slangy expressions here--that would just get out of control. Maybe another day.They're in no sensible order whatsoever.If they have a link, I've already discussed them in more detail--click to see.Some corrections have been made (in green) since comments started coming in. Please see comments for more discussion of those...
BrEAmEmobile (phone)cell (phone), cellular phone engagedbusy*directory enquiries   directory assistance (aka information)

telephone directory  phone book*

service provider  carrier

answerphone /Ansaphone [answering machine]

dialling code   area code

bleeper  beeper (pager in both dialects too)

phone box, telephone call-box  (tele)phone booth

reverse-charge call   collect call*

dialling tone   dial tone

ex-directory   unlisted (of a person/telephone number)

freephone number (0800 number)   toll-free number (800 number) hash keypound keytelesalestelephone soliciting (telemarketing in both too) push-button phone*Touchtone phone*3G WAP0898 number900 number (premium in both) 1471 (pron. one-four-seven-one; identifies last caller)*69 (pron. star-sixty-nine; call-return)

The list credits: Thanks to the following people for suggesting some of the above differences: Mark Allen, Philip Nelkon, and Ofer at Tomedes). And to the OED and Better Half for confirming some. 
The other thing to mention here is the difference in verbs of telephoning, particularly BrE is to ring someone, to ring someone up or to give someone a ring. In AmE, one can use call in all of these cases. While call is not just AmE in this case, it is stereotypically American--so much so that I've taught myself to say phone, which is shared by both dialects and makes me feel less self-consciously American while not feeling like I'm in a Jeeves and Wooster novel. I don't know why saying ring makes me feel self-conscious when I've easily adapted to lots of other BrE words. Perhaps verbs are harder to make oneself at home in.

Why are there so many differences? That's relatively simple: the technology was introduced after these dialectal groups were well and truly separated--so, if a new word for something needs to be made up in one country, there's no reason why the other country should come up with the same word. In some cases, the technologies themselves took different paths. Similarly, (BrE) motorcars/(AmE) automobiles and road systems have hugely different vocabularies (click on the transportation tag for some--but I've yet to do the Big List of Car Parts).

A few other differences to mention:
(1) The sounds that phones make are different in different countries. When I first moved to South Africa, I mistakenly believed that everyone I tried to ring/call was on the phone, because the ring tone to me sounded more like the American busy/engaged signal than the 'ringing' sound. (I've also been tempted to think, in various countries, that the phone is broken--because the dial[ling] tone sounds 'angrier' than the American one.)
(2) The US and Canada share the country code '1' (hey, they started this whole telephone thing). Historically, one dial(l)ed the '1' to let the telephone exchange know that an area code was the next thing coming--and one still does have to prefix the number with '1' whenever one dials out of one's own area code. That evolved into a North American country code, when such things became relevant. In many other countries (including all of Europe that I've telephoned in and South Africa), when dial(l)ing a non-local number, the first thing you dial is '0'. But whereas the '1' is not represented as part of the area code in the US (it's separated from it by a dash), the '0' is represented as part of the dial(l)ing code in the UK. Here are examples of each, using government tax assistance numbers in each case, as they are presented on the agencies' web pages:
US: 1-800-829-1040
UK: 0845 300 3900The tricky thing for USers to learn is that the 0 at the front of a UK-style number needs to go away when you dial from outside the country. So, if you wanted to phone the UK number from abroad, it would be:
+44 845 300 3900And before the country code (44), one needs to dial the international access code, which has been 00 in every other country I've used a phone in, but is 011 in US (and Canada too?).  Another thing that surprises North Americans abroad is that in other countries, all the phone numbers don't have to have the same number of digits. For example, the London codes 0207 and 0208 are shorter than my city's code, 01273. And until a few years ago, they were even shorter (020).

Which is all to say that if you live in North America, you have a lot to learn about how telephones work when you go abroad. But if you live in the UK, you can travel a lot of places and still apply the same telephonic logic to the new country's phone numbers. Unless you're travel(l)ing to North America, of course.