1.   be connected to :把...连上，例如，

Please connect the computer to the Internet.
Can be connected to the printer and print data.可以连接打印机并打印数据。
2.be connected with :把...与...连接起来。例如，

It serves as a bridge which connects the mainland with the rest of the world.(它是连接祖国大陆与世界其它地方的桥梁）。 The mainland is connected with the rest of the world.
I always considered your brother to be connected with that club.
我原来总认为你兄弟和那个俱乐部有关系。
3.be connected by：被...连接在一起。例如，

The whole world is connected with by computers and it is becoming smaller and smaller.(整个世界被电脑连在了一起，世界变得越来越小了）.

区别：to ,with均表示连接的对象。可通用，但是，with含相互联系，而to含一方向另一方，有可能只是单向的联系。
by后面接方式，并不接对象。
My computer was connected with yours.--我们的电脑相联
My computer was connected to yours.我的电脑联上你的电脑。
My computer and yours were connected by a wire.--我们的电脑是通过一根网线相联的。