They are not the same thing.
A bug is unintended behavior of a piece of software: the software doesn't do what it is supposed to do. Bugs can live at all levels of software development, ranging from plain old typos through logical errors up to inadequate functional specs.
An exception, by contrast, can refer to either an unusual condition of a program, deviating from normal operation, or, more specifically, to the language construct used to signal and handle such conditions.
The fact that an exception occurs can be a sign of a bug, but often it isn't. For example, an application that is supposed to download a document from a URL and process it locally might throw an exception when the remote server is down: the application is deviating from normal operation (it cannot download and process the document), but if it handles the exception properly and recovers, then there is no bug.
Conversely, the presence of a bug doesn't necessarily manifest itself as an exception. An application might silently discard data you enter instead of storing it in its database; no exception gets thrown, but it is still a bug.