My buddy Isaac just bought a GX1 -- in fact, he bought two of them.   This gave us a great chance to compare lenses:  The non-zoom, 20mm 1.7f lens (about $350 on Amazon) with the power zoom, 14-42mm 3.5f lens (about $325 on Amazon).  There's also a very interesting third lens to consider, which we didn't specifically test:  The non-zoom, 14mm 2.5f lens ($286 on Amazon).

Here's what Isaac had to say about the 20mm lens:

 

• Comparison: Lumix 14mm vs Lumix 20mm pancake lenses
  
• Lumix G 20mm f/1.7 pancake (this is the lens Isaac got)
  
• Panasonic Lumix G 14mm f/2.5 (not as low of an f-stop, but very wide angle)
  

As Isaac mentioned, the lower the focal length, the more distortion you'll see when taking close-up portraits of people.  @m43photo has a GREAT article describing that phenomenon here.  The lower the focal length (i.e., "14mm" vs "20mm") the wider the angle the lens has.

And Isaac is right -- the 20mm lens does focus slowly, which makes it hard to take pics of a child that's often moving.  However, when it captures a picture, the shot is awesome.  Here are a few great pics I got in just a few minutes of shooting:

I found the best strategy to be just keeping a fixed manual focus, and then working to keep the camera the correct distance from the moving subject.

Speaking of manual focus, there's a nice big manual focus ring on the 20mm lens, but I found it didn't focus as quickly as I'd like.  It took 3 full turns of the ring to go to each extreme.  From what @m43photo said in his blogs, the 14mm lens may focus more quickly in manual mode, so I'm super interested in comparing it with these lenses, which I hope to do at some future date.

In sum, I'd say that if you want to take some killer pics, go with the 20mm lens.  The 1.7f is just awesome (note the super shallow depth of field in the two pics above, meaning that the backgrounds are very blurry), and the lens is compact.

If you want more of a "do it all" lens and you're willing to live with a power zoom and power focus (i.e., no manual controls), then go with the also very-compact 14-42mm zoom lens.