Is there any reason why fluorescence spectroscopy is said to be more sensitive than Uv-Vis?
Fluorescence is a 'zero background' or absolute type of measurement meaning that single photons can be measured against a 'dark' background so the sensitivity is huge, and limited by the fraction of light caught and efficiency of the detector.
Absorption is a relative measurement as it is the difference in two large numbers, one the light intensity with no sample $I_0$ and the other when the sample is present I. In terms of photons these intensities are huge, say $I_0=10^6$ and a 1% change would reduce the number of photons to $10^6-10^4$; still a large number and so the absorption measurement is looking for the small difference in two large numbers. This measurement is limited by noise in detectors and other electronics. (Its a bit like trying to weigh the captain of a ship by weighing the ship with him standing on the deck and then when he is not there.)
In fluorescence, what is detected is really photons emitted. The measure is done in absolute term (versus 0 , the dark) and the response can be easily amplified.
Conversely, in absorption is a difference that must be measured. And this can of course be arbitrarily tiny.
It is like massing a sample. Is easy to weight a mg than detecting a mg difference among two " about 10 kg " samples.
This general consideration is exacerbated by other practical details, as listed in the answer by Dan.
Also your consideration about geometry applies, which again is dictated by the above.
Edit: Porphyrin answers appeared after I submitted this. Else I wouldn't have answered as this is basically a duplicate.
Absorbance may occur from non-analytes, while fluorescence is often quite specific to an analyte or a fluorophore-tagged analyte. Also, instrumental setup could play an effect. UV-Vis specs are often horizontal in terms of incident beams, sample, and the detector. Fluorescence specs are often at 90 degrees to "extract" the fluorescence from the sample to the detector. If you are only "extracting" what you are interested in, I'd imagine the S/N would be better, yielding higher sensitivity.