'Giclee' is just a fancy way of saying "that which erupts from an inkjet nozzle" and was coined by Jack Duganne so he could avoid calling his fine art prints 'computer generated' or 'inkjet' prints. With such printing a dye or pigment is sprayed upon a paper to create an image.

In chromogenic printing nothing is sprayed upon the paper. Rather, the paper has three light sensitive layers that are exposed to light projected through a negative or, in the digital age, projected onto the unexposed paper using RGB lasers or LEDs. The latent image on the exposed photosensitive paper is then chemically developed so that the dye couplers in each layer react to the exposed silver halide in each layer to form color dyes. Chromogenic printing is how the vast majority of color images produced in the film era were created.

Which process for making prints is better than the other is a matter of taste and how one wishes the print to appear. It is a little more complicated than the simplistic idea that inkjets gives finer control over subtle differences in color and photosensitive papers give smoother transitions that appear to have higher effective resolution. But the gist of what your printers are telling you is true - it's a matter of taste and deciding which tradeoff is more important to you for a particular image.