Tube or tape being pushed past it's limits is generally what saturation plugins try to emulate. This will affect the shape of the waveform and level off any transient peaks, but because saturation tends to emulate analogue gear it won't completely bend your waveform out of shape like distortion might. See the below image for a visual description of how soft clipping in analogue gear works, and consequently how saturation plugins tend to work. The bottom image is saturation or soft clipping, the top image is distortion or hard clipping.
So where can we use saturation? There are many places where it works well. Important to note here is what saturation does to the frequency content of a signal. Think about synthesis, when we add odd or even harmonics to a waveform, it changes in shape. If saturation is changing the shape of a waveform, logically we are therefore adding odd and even harmonics to the sound. Harmonics are whole number multiples of the fundamental frequency of a sound, so in short what we are doing is adding frequency content to the upper part of the sound. This has the effect of brightening it, and changing the sound's tonal characteristics. This can work well on almost any sound, but I particularly like the effect saturation has on drums and vocals. On drums it will have the effect of levelling the peaks of the drum sound. This can make the drums sound fatter and punchier in some instances, and acts in a similar way to a compressor in this regards, as it also allows you to bring the overall level of the drums up in the mix. However unlike a compressor it will also affect the tonal characteristics of the sound, which may be a good or a bad thing depending on the circumstances.
As I mentioned before distortion is a more extreme version of saturation and will rectify a waveform or bend it's shape much more than saturation will (see the top of the above image). This tends to add in much more harmonic content and affects the frequency response a lot more than saturation will. It also adds in more odd harmonics that saturation does, resulting in a squarer waveform and a more brittle and harsh sound. Distortion has it's place too, and again can sound good on drums, of course on guitars in the form of overdrive and such, and a variety of other material to get a generally more aggressive sound.
One thing worth mentioning is that there is a another type of distortion which works on a different principle, and thats bitcrushing. This reduces the dynamic range of a sound and therefore basically reduces it's quality. It does this by reducing the number of binary digits (bits) that the computer can use to represent different loudness levels at different times. Many bitcrushers also have a function to correspondingly reduce the sample rate, which affects the frequency response of a particular piece of audio (see below picture) the combination of which will distort the sound in a very particular way. Bitcrushers generally have a very harsh sound, much more so than saturation and that only the harshest of distortion can match. Personally I find using a bitcrusher in parallel on my drums and lead synths can be a great way to add loudness and bite. Used subtly it can also emulate the sound of old hardware such as an MPC60 which was not able to reproduce a large bit depth or sample rate when playing back sounds. This can help give a tastefully lo-fi sound to your production.