To clarify a little - I've been looking at the history of US Presidents and British Prime Ministers. What strikes me is the amount of times an assassination has been attempted (and succeeded) against the US President when compared to the Prime Minister. I'm interested what the historical reasoning for this is. The title sounds like I'm asking what the assassins motive was, this is not the case. I'm more interested in why assassination even becomes an option.
I would propose that prime ministers just aren't that important. They don't symbolize a country in the same way a president does. The American presidency is a cultural history subject on its own, whereas in Britain there isn't really a PM cultural history to the same degree. This is probably because there's a royal family that is much more symbolic and culturally relevant to the country's history.
As far as assassination, it makes a statement. Because the American presidency has such a cultural history and symbolism, if you stand against what the president stands for killing him feels like killing a tyrant (see John Wilkes Booth). People assassinate leaders because they disagree with them on a very deep level (i.e. believe themselves on the opposite side of the fight) or because they are nuts. You aren't going to be able to kill a president with subtlety and pretty much anyone who wants the president dead is going to want to ensure a method that has a reasonable chance of success. So that's why guns are most common in American assassinations.