Musical tones are pitched and tuned to a musical scale. They come from any number of musical instruments or the human voice, and can be combined to form harmonies.
Noise is unpitched. The frequency might be extremely low or high. Noise might be natural, mechanical or electronic. The frequencies are disorganized and not harmonic.
Harmonics are overtones that add character to a tone. The pitch is easily perceived, but it may be complex, like a piano tone or fuzz guitar. Harmonics preserve the pitch, and the tone remains musical.
Overtones can be non-harmonic. They have a complicated mathematical relation to the pitch, and confuse the pitch. Non-harmonic overtones tend to make a tone noisy. Gongs and cymbals are examples of non-harmonic overtones, and are noisier than the pure tone from a triangle, for example.
White noise sounds like radio static, a "hissy" collection of random signals. Thunder, wind and surf are all variations of white noise.