Twitter widget appears to be broken: "Application not installed"
I had a similar problem with another app (not Twitter) I went into my app manager and saw that the app was installed on my "External Storage" even if I never moved it there. I moved it back to the Internal Storage and it was fixed. My card was already fine to begin with but for some reason, the widget didn't like it. 
Anyway, I never recommend putting apps on the SD card, it slowed my S3 to a crawl. Keep it for media and docs.