A neon light is the sort of light you see used in advertising signs. These signs are made of long, narrow glass tubes, and these tubes are often bent into all sorts of shapes. The tube of a neon light can spell out a word, for example. These tubes emit light in different colors.
A fluorescent light, on the other hand, is most often a long, straight tube that produces white light. You see fluorescent lights in offices, stores and some home fixtures.
The idea behind a neon light is simple. Inside the glass tube there is a gas like neon, argon or krypton at low pressure. At both ends of the tube there are metal electrodes. When you apply a high voltage to the electrodes, the neon gas ionizes, and electrons flow through the gas. These electrons excite the neon atoms and cause them to emit light that we can see. Neon emits red light when energized in this way. Other gases emit other colors.
A fluorescent light works on a similar idea but it has an extra step. Inside a fluorescent light is low-pressure mercury vapor. When ionized, mercury vapor emits ultraviolet light. Human eyes are not sensitive to ultraviolet light (although human skin is -- see How Sunburns and Sun Tans Work!). Therefore, the inside of a fluorescent light is coated with a phosphor. A phosphor is a substance that can accept energy in one form (for example, energy from a high-speed electron as in a TV tube -- see How Television Works) and emit the energy in the form of visible light. In a fluorescent lamp, the phosphor accepts the energy of ultraviolet photons and emits visible photons.
The light we see from a fluorescent tube is the light given off by the phosphor that coats the inside of the tube (the phosphor fluoresces when energized, hence the name). The light of a neon tube is the colored light that the neon atoms give off directly.