From my limited knowledge, Lab tools works in L*a*b* color space and the corresponding tools and curves in Exposure tab works in RGB color space. Lab color space separates luminance from chroma, so when adjusting lightness/contrast with Lab tools, the color is not affected. On the other hand, when you try to adjust exposure with RGB tools/curves, the saturation of colors will be affected. Adjusting Lab chrominance should have no effect on exposure but adjusting RGB saturation will affect lightness. The RawPedia should have a good explanation of the difference of both tools:
In practice I found that adjusting exposure with Lab tools will produce “dull” color and RGB tool doesn’t do that. I would just use lab adjustments when I think the saturation of the image is a little bit too much, or if I just want to keep current saturation “as is”. My 2 cents.
I’m going to diverge from @syyrmb 's opinion here. For my way of editing, I prefer working in the L-a-b space. You can adjust the lightness, chromaticity and hue without affecting each other. And in RT, you can affect those three components on individual channels. I created a short, Quick Tips video that demonstrates this. I hope it helps.
I haven’t found it to be the case that Lab adjustments necessarily produce “dull” colors. The chromaticity control certainly lets me “overdo” colors if I’m so inclined (although, a gentleman refrains).
Oh sorry, I forgot the chomaticity tool here. I was talking about lightness and contrast tool since they were my most often used Lab controls and they produce “duller” color compared to their RGB counterparts. The chomaticity tool, not to much, and it doesn’t cause hue shift like the Saturation tool when “overdo” it. I totally excluded the chomaticity tool when saying that…
I use the LH, CH and LC controls in the LAB tool. Lightness, Contrast and Chromaticity, and the tone curve, I use the exposure tool.
I will test the use of the Lightness, Contrast and Chromaticity controls, and the L* curve instead of using these controls in exposure to see the differences I get and which one works best.