In everyday usage, they refer to the same thing, but technically they refer to different things. Toilet is the actual equipment you use to "do your business" (i.e., to urinate or defecate). Bathroom literally means the whole room, in which there is a toilet, a bathtub, and a sink. Restroom is generally used for public spaces (such as restrooms in a restaurant or a rest stop along the freeway.) However, because as human beings we tend not to like to talk about our bodily functions, we generally use euphemisms so that we don't have to say "urinate" or "defecate" directly, especially in polite company. That is why we usually say "I have to go to the bathroom," or "I have to go to the restroom," and slightly less often, "I have to use the toilet."

They are only the same in countries where it is common to have the toilet in the bathroom. A bathroom is a room in a house with a bath, or more likely these days - only a shower. In the USA, and some other countries, that is where the toilet is as well. A toilet is (as Brian said) the physical equipment, and/or the room it is in. A washroom is a room - usually in a public place, that has toilets and handbasins (for washing). It would almost never have a bath, and rarely even showers. If you ask for the bathroom in a country where it is uncommon to have the toilet in the bathroom, you are likely to come back sheepishly asking for the toilet.

They are all the same.