Dependency Injection vs Factory Pattern

The best definition I've found so far is one by James Shore:

"Dependency Injection" is a 25-dollar term for a 5-cent concept. [...] Dependency injection means giving an object its instance variables. [...].

There is an article by Martin Fowler that may prove useful, too.

Dependency injection is basically providing the objects that an object needs (its dependencies) instead of having it construct them itself. It's a very useful technique for testing, since it allows dependencies to be mocked or stubbed out.

Dependencies can be injected into objects by many means (such as constructor injection or setter injection). One can even use specialized dependency injection frameworks (e.g. Spring) to do that, but they certainly aren't required. You don't need those frameworks to have dependency injection. Instantiating and passing objects (dependencies) explicitly is just as good an injection as injection by framework.

What makes you say that either a singleton or a static method isn't thread-safe? Usually both should be implemented to be thread-safe.

The big difference between a singleton and a bunch of static methods is that singletons can implement interfaces (or derive from useful base classes, although that's less common, in my experience), so you can pass around the singleton as if it were "just another" implementation.