You ask what makes Britain's weather so unusual, but what you really want to know is what makes Britain's climate unusual. Climate refers to the typical conditions: weather is the day-to-day change in those conditions. So when we're talking about the fact that the Amazon is always hot and wet, and that Antarctica is generally very cold and dry, we're talking about the climate of those places. When we talk about the fact that it was windy, but didn't rain today, that's weather.
So doing a web-search on "UK weather" will give you lots of pages telling you what the UK is like today, or a few days from now -- not what you want. Doing a web search on "UK climate" might give you the information you need.
In particular, the UK Met Office has a good set of web pages on the climate of various regions of the UK, along with some hints as to why those regions have that kind of climate. Choose a region from the "Browse Past Weather By Location" bar at the top of the page to learn more.
Which brings up two important points. First, you say that Britain's weather is unusual. Britain doesn't have one single climate. Temperature, wind, and rainfall change quite a lot in the various parts of the island, as you'll learn from the web page above. Second, unusual compared to what? In particular, some parts of the northwest U.S. and British Columbia (in Canada) have climates similar to England and Wales. Look at a map of the world. What do England and Wales have in common with British Columbia? Thinking about that might help you answer your question.
Try the links in the MadSci Library for more information on Earth Sciences.