Twitter requires a Secure Socket Layer (SSL) connection (https rather than http). The Arduino ethernet and wifi libraries do not support that protocol. I think it is a matter of memory. The larger devices that Arduino has in the works may have enough to do that eventually.
Or, you can use the RFXduino gateway system
which allows SSL connections from Arduinos. Even has a "Send a tweet to Twitter via https" routine already built-in to the library (see doc pages
for code snippets.)
WiFi from an Arduino is actually easy, as well as inexpensive, if you take the right approach.