This video showcases the spectrum produced by three different types of lighting.
A tungsten filament glows because of electromagnetic radiation generated by the thermal motion (more specifically, acceleration) of charged particles. Production of light in this manner is called thermal radiation or incandescence. Since there is a continuous range of thermal motion, incandescence produces a continuous spectrum.
LED is a p-n junction that emits photons when conduction band electrons recombine with valence band holes. The energy (and thus wavelength) of the photon is equal to the energy transition made by the electron. This is kind of similar to how light is produced in a gas discharge lamp. However, unlike a gas atom where transitions are between discrete energy lines, in the p-n junction the transitions are between the conduction band and valence band. The spectrum of a LED light is thus not discrete, but centred about the band gap energy.
A fluorescent tube is like a improved mercury discharge tube. Unlike a mercury vapour lamp where a lot of energy is wasted in the ultra-violet light produced, a fluorescent tube is coated with a fluorescent coating. The fluorescent molecules are excited by ultra-violet photon from ground state to one of the vibrational states in the excited states. When they de-excite (through collisions with other molecules), they emit a few visible photons as they cascade from the vibrational states. As a result, the spectrum of a fluorescent lamp consist of both the discrete lines due to the mercury vapour and a continuous spectrum due to fluorescence. The choice of fluorescent material gives rise to different flavors such as warm white, cool white, daylight, etc.