Wow, this thread is all over the place. If you're bringing up something similar but not exactly the same please start a new thread. It's VERY hard to answer multiple problems in one thread.
NYG8OR: Do scheduled posts work for you? Try scheduling a post to publish in the future and see if it publishes when it should. The requests to Twitter are scheduled in much the same way.
WQ32: The plugin DOES cache the results from Twitter. It stores the response as an option and stores a second option with an expire date/time. It will continue to use the results forever, but once the time has expired it will try to update that option with new results (on failure to update though, it will continue to use the old data).
Dan Davies: It's likely that something else has changed. Maybe a security update on your server closed off self-requests (where the server requests a page from itself)? These are used to trigger the update process that grabs fresh data from Twitter. As for "many people" having this problem, over 14,000 people have downloaded the plugin since the new version was released. It looks like maybe 10 have a problem, so that's less than a tenth of a percent...which I actually think is quite good.
firstname.lastname@example.org: Try checking your API limits (WP-Admin->Settings->Twitter Widget) and testing scheduled posts as mentioned above.
marlonlamancio: "Does not work" isn't really enough of a description to work off. Please start a new support thread with as much information as you can (what specifically does it do? Were you able to add the widget to a sidebar? What settings are you using? etc)
panmoria: Try checking your API limits (WP-Admin->Settings->Twitter Widget) and testing scheduled posts as mentioned above. Also, I'll re-iterate what I said to Dan. It looks like maybe as many as 10 people are having this problem, but more than 14,000 have downloaded and installed the new version. Less than 1/10th of one percent doesn't seem like "many people". I'm not trying to be a pain, just trying to keep things in perspective.