A valve and a tube are the same thing -- "tube" was originally more common in American English, while "valve" was more common in England. A "vacuum tube" is any one of a number of types of electronic component based around an arrangement of electrodes inside a sealed glass unit, from which all the air has been removed.
The kind of tube used in amplification is a triode. You give it a steady high voltage on one input, and a varying low voltage input (like the output of a guitar or a record needle) on the other. The output is the high voltage, varied according to the low voltage signal -- that's amplification.
This is the same behaviour you get in a transistor, which is smaller, less fragile and easier to manufacture in high volumes.
The interesting thing about valves is what happens when the signal input is too high.
For a theoretically perfect amp,
output = c * input where
c is a constant. A tube amp will approximate this for most of its range. But what happens if
c * input is greater than the maximum output of the amplifier?
You might imagine that the output is either
c * input or
max_output, whichever is lower. If that happens, you get "hard clipping" as shown by the red lines in this graph.
Tubes, though, don't respond quite so harshly. As you approach the maximum, the amplification ratio gradually eases off, so instead of just cutting off the top of the wave, you get a gentle curve, described as "soft clipping".
When used for music, the outcome is a distortion effect that people tend to describe as "warm". By contrast, transistors tend to have a sharper curve, which people tend to describe as "harsh" -- although there are transistors made specifically for music which have a more musical sound.
Hence an amplifier made with valves is considered to add an appealing sound when pushed a little, and it also provides a safety net when mixing -- occasional peaks are handled gracefully.
A tube/valve amp VST, as you can imagine, simulates those response curves, to create a sound that's close to what a valve/tube amp would produce.
There is a certain element of familiarity here. All the classic rock/pop recordings were made with tube pre-amps, and so we have learned to love the effect.
Tape saturation is a similar effect, but with a different history.
When audio is recorded onto analogue magnetic tape, the varying electrical signal is passed to an electromagnet, which creates a varying magnetic field, which leaves its mark on a moving tape.
Again, the interesting stuff happens when the input signal is too high for accurate reproduction. Again, the way magnetic tape handles an excess signal is to flatten the peaks gently.
The effect is so well loved that there are high end studio accessories containing a short loop of tape, a write head and a read head, so that the performance being recorded gets real tape compression applied before it is digitised and recorded to hard disk.
Tape also has a non-linear frequency response, so can be compared to a certain set of EQ settings.
Again, a VST simulates these effects.
Both tube and tape, when overloaded, are kinds of compression. They reduce the difference between quiet parts and load parts -- in this case by amplifying the loud parts less, and doing so in a way that distorts the sound in a manner that we have learned to enjoy.
What do you do with this knowledge? Well, try them all and use your ears. If you like what you hear, it's good. If you don't, it's bad. If you can't tell the difference, it's pointless.
As with all mixing and mastering, it's a minefield of superstition and psychosomatic perceptions -- and you might not have the "golden ears" to tell the difference; many of us don't. If you really care, do blind A/B tests. Or, just go with what sounds good to you, psychosomatic or not.