The most important thing about げ is that it describes an observed quality. That is, you cannot use げ to refer to yourself:
○ 毎週楽しく聴かせていただいています。 I enjoy listening every week.
× 毎週楽しげに聴かせていただいています。 (incorrect)
The reason for this is that げ (which in kanji would be 気, but it's never written in kanji) is defined as そうだ or らしいようす, according to Daijisen. (Daijirin also lists 気配 as a definition, and you can see my answer to this question about 気配 to get a sense for that word.) Since そう is for describing something based on observation, it would be odd to use it to describe something about yourself. But you can easily use it to describe someone else:
○ 少女はうれしげにプレゼントをあけた。 The girl happily opened the present.
○ 少女はうれしそうにプレゼントをあけた。 (equivalent)
Why not just use the normal adverbial form of うれしい (うれしく) here? Because うれしく would presume that you have accurate knowledge of someone else's emotional state, which you don't (unless you're writing a book and talking about one of your characters). All you have are observations, and so you have to show that your judgment of うれしい comes from observation by using either そう or げ.
Although you could technically use げ with any word fit for describing behavior, not every such construction is common. For instance:
○ みんなはケーキをおいしそうに食べている。 Everyone looks like they're enjoying the cake.
？みんなはケーキをおいしげに食べている。 (same meaning, but far less common)
When you can freely substitute げ for そう (as with the うれしい example above), I don't personally feel any difference in emphasis, but a native speaker or someone with a solid reference on this may weigh in with a different opinion.