The instrument is decoupled from the telescope. This is indeed the main advantage for telescopes that do not allow heavy loads. The design constraints like stiffness, weight and volume are highly relaxed.
Spectral stability. One of the most important sources of error in spectrographs attached to telescopes is the shift of the spectrum on the detector due to mechanical flexures along the observations. With fibre optics, the spectra become much more mechanically stable. Residual shifts are ultimately related to changes in temperature, atmospheric pressure, humidity and air turbulences. An example of a high stable high radial velocity accuracy spectrograph is HARPS
Versatility and handling
Multi-object spectroscopy. Fibres may allow simultaneous observations of many objects with only one spectrograph. A number of fibres can be aligned in a row and placed in front of the spectrograph slit. On the other end, each fibre may be mechanically placed in front of the stellar target at the telescope focal plane. The CCD can therefore record the spectra from each fibre. FLAMES MOS spectrograph is an example
Integral field spectroscopy. A bundle of packed fibres may be placed in front of the image of an extended object like a galaxy. The output fibre ends are arranged along the slit of the spectrograph. In this way, spectra of portions of the target can be simultaneously recorded and get a spectral map of the target. VIMOS instrument in Paranal Observatory includes a huge fibre Integral Field Unit (IFU)
Image or pupil slicer. Like in the case of the integral field spectroscopy, a bundle of fibres may be used to increase the resolving power of the spectrograph. Instead to collect the flux of the image of the star with a single thick fibre, a number of compacted fibres with small core diameter may be used. In this case the equivalent slit width is smaller and therefore the resolving power is bigger. The same principle may be applied to “slide” the pupil of the telescope
Field acquisition and/or telescope guiding. In multi-object spectroscopy a big number of individual fibres are accurately placed on plates located in the focal plane of the telescope to pick up the light of individual stars. The fibre output ends are arranged on the slit of the spectrograph. In this case it is possible to take spectra of many stars simultaneously. In order to accurately align the stars on the fibre input ends, the support plate includes few coherent fibre bundles placed on the image of reference stars. The fibre outputs are assembled on a monitor camera. The telescope is moved in order to centre the reference stars simultaneously of the coherent bundles. In this situation all the other fibres are in front of the stars to be analyzed.
Higher spectral resolution. The resolving power of the spectrograph is slightly higher for a circular aperture than for a slit with a width equivalent to the diameter of the fibre. However this resolution gain is at expenses of flux reduction (see Section 5.4.3)
Photometrical scrambling. For extreme high precision measurements of the radial velocities (e.g. detection of exoplanets) optical fibres increase the photometrical stability of the point spread function on the detector. Additional information may be found in this paper
Interferometry. Single mode fibres are commonly used in astronomical interferometry to combine coherently the beams from 2 or several telescopes. Fibres reduce considerably the number of optical elements in interferometry designs. AMBER VLTI instrument uses SM fibres
Laser Guide Star devices. The fibres are extremely helpful to launch high power continuous wave lasers as in the case of the so-called Laser Guide Star launchers. A description of the ESO LGS facility can be found here