At the risk of sounding like a skipping LP, ultimately you need to play with a GF1 to really know if it will work better for you or not. And recommendations are hard without knowing how you use the camera.
I don't own a D60, but I've played with a lot of Nikon DSLRs, and I think many of the differences are in a real sense more species-specific than model-specific.
have been shooting with a Nikon D60 and 35mm f1.8...Recently the GF1 with the 20mm lens has started looking attractive...what I will lose and gain performace wise?
The GF1 w/20 is, I think, significantly smaller than a D60 w/35. Some will argue this point. If you never add another lens the difference isn't huge, but it is significant. Less weight around your neck, smaller in the hand, fits into a smaller bag. Any lens you add will make the difference greater (compared to similar-spec APS lenses).
The difference between Live View mirrorless and DSLR OVF viewing/composing is also pretty significant. Live View lets you see exposure and white balance effects in real time, pre-exposure. Auto-gain makes low light viewing and dof checking easier. Precise framing from low and high angles is possible. Real usable Live View was a revelation for me, and I don't think DLSRs have any advantage here.
That said, I need an EVF for outdoor use, and the GF1 EVF is adequate, but not great. It will also add $100+ to the overall cost. I prefer cameras like the Pany G2 with onboard EVF.
Your D60 probably has slightly less noise at high ISO, but it's an older sensor and the difference isn't as great compared to a D5000 or D90. I think you may be looking at a 1/2-stop advantage. If that much difference is important to you, maybe you're better off looking at the GH2 or a D5000.
I'm not sure what you'll find comparing AF speed. The GF1 AF system is more advanced, but the D60 PDAF might be a bit faster in some situations than the GF1 CDAF. Your D60 is probably a bit better for action/sports, but you're generally not shooting much of that with a 35/1.8.
The GF1 will shoot HD video, it can use hundreds of legacy MF lenses with affordable adapters, it will weigh less around your neck and fit into a smaller bag. You may find live view a better way to take photos. Whether you'll like it better than a D60, or a newer DSLR, you've got to figure out for yourself.