If you don't know what flux throughput is, there is little change you would understand an answer about flux throughput advantage.
For any spectrograph (or any measurement), you have to be concerned about the ratio of the signal versus the noise. For a spectrograph, the signal is easy to understand. The more light you collect, the stronger the signal.
One source of noise is the electronics of your detector. For astronomical devices, the detector is usually cooled which significantly reduces noise from the electronics.
A spectrograph sorts photons by wavelength. You parse light into different buckets that cover a range of wavelengths. This isn't a perfect process and some photons don't get sorted properly. This is known as stray light - that is to say light that is scattered inside the instrument. A properly designed spectrograph will have low stray light. There are very efficient methods of correcting for stray light in a spectrograph see https://www.researchgate.net/publication/7254062_A_simple_stray-light_correction_method_for_array_spectroradiometers
You also have to consider the resolution. Detecting the signal in a 1nm bucket gives you a strong signal. Parse that light into 0.1 nm (1 angstrom) or 0.01 nm (0.1 angstrom) bins spreads that light out even more and the amount of light in each bin is lower. (It is also more difficult to sort the light so precisely).
Early spectrographs used a diffraction grating to sort light. A diffraction grating reflects (or refracts) light at different angles depending on its wavelength. A longer focal length increases the resolution, but may increase stray light. A longer focal length also increases the physical size of a spectrograph.
BTW: Cassegrain is a specific
Using a second diffraction grating (double monochromator) improves resolution since you are basically sorting the light twice.
A spectrograph can use a single detector and rotate the diffraction grating(s) to measure the signal vs wavelength. The slower the scan speed, the higher the resolution. This also means more time to collect a spectrum. Virtually all spectrographs these days use detector arrays which allows measurements of light at thousands of wavelengths simultaneously.
I hope this background helps you enough to let you find additional answers on your own.
BTW: Cassegrain is a specific type of optical design for a telescope. Coude (French for "elbow") refers to a focal point. The coude focus is stationary which makes it easier to install large or complex equipment.