How do these two differ, for example:
寂しそう vs 寂しげ
楽しそう vs 楽しげ
言いたそう vs 言いたげ
大人げ vs 大人っぽい（...？ Not sure if this one works.)
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They are the same ("seems like") but 〜げ has more of a connotation of 「それらしい」 or 「っぽい」with the げ coming from the character 気 as in 気分. I remember it as "that sort of feeling".
Arguably this makes 〜げ more subjective whereas 〜そう is more objective but only so far as the observation is shared with others in the same/similar view point as the speaker.
A word about usage:- You commonly use 〜げ when expressing your feelings. It comes from the kanji 気, so this serves as a reminder. It matters "what" you are talking about, not just that an adjective comes before it.
So observing the rule about expressing your feelings.
1A. 寂しそうな顔をしている = OK (with a lonely face)
1B. 寂しげな顔をしている = OK
2A. 楽しそうな雰囲気 = OK (seemingly enjoyable atmosphere)
2B. 楽しげな雰囲気 = OK
3A. 言いたそう = OK (with a desire to say ...)
3B. 言いたげ = OK (NB: this is common, but not widely used with other verbs)
4A. 大人っぽい = OK
4B. 大人げ = (corrected: this is ok as in 大人がある or 大人げない or
Alternatively these can be followed by に, as in 言いたげに見ている (looking with an expression of wanting to say something).
Origins:- Apparently this word has been around for sometime in certain parts of Japan (Ref: http://otasuke.goo-net.com/qa5447963.html?order=DESC&by=datetime > Answer 2), but some consider it to be 若者言葉 that is making the transition to accepted vocabulary (Ref: http://ja.wikipedia.org/wiki/%E8%8B%A5%E8%80%85%E8%A8%80%E8%91%890 + own experience)
Well, basically ADJ-I's radical + げ means some sort of "with adj-as-a-noun"; it turns your adjective into a noun, expresses appearance. To make myself clear, let's see a few examples taken from the net:
A bar in Roppongi with a somehow different enjoyable atmosphere
the artists walking the red carpet sigh with a sad look.
Even though I've never encountered it, I infer that "言いたげ" means "with the apparent wish to say" (since the base was "言いたい", "I want to say"). In both cases, this is clearly something that you are observing. I think that you will mostly encounter げ+に or げ+な.
そう, on the other hand, is trickier. It's either your impression, or something you've heard.
Tomorrow, it will rain (according to the weather forecast I heard)
Tomorrow, it looks like it is going to rain (that's my feeling, I might be wrong).
He seems happy, he's happy.
I heard he's happy.
I think he's trying to say something. (nota: I don't think it's natural, I have almost never heard this kind of sentence).
I heard he wants to say something.
For the last one, I'd say it doesn't work, since you're working with a noun. And appearance or hearsays with nouns is another topic…