This one is a commonly asked question, but luckily, has a simple answer. The short of it is this: skewT's have straight isobars, while tephigrams have straight dry adiabats. As for the long of it...
Skew-T log-P diagrams are descriptively named. Temperature, T, is skewed to about a 45° angle. Pressure, P, is set to a log scale. Simple enough – once you get past how scary the diagram initially looks.
Skewing the temperature lines in this way improves clarity, separating out the temperature and dew-point temperature a little, and orienting the temperature lines such that they progress upwards, rather than off to one side. And pressure decreases exponentially with height anyway, so using a log scale is the logical choice; this way, altitude can increase linearly.
The tephigram, however, doesn't base its orientation on the pressure lines at all. Instead, it uses entropy.
You see, if you were to take a parcel of air and lift it vertically, it would cool as the pressure surrounding it decreased. The thing is, that parcel can return to its original temperature again if you bring it back to where it started.
In other words, the entropy of the parcel doesn't change – provided the system is isolated. This is known as an adiabatic process, and so the lines on the tephigram for which entropy is constant are known as 'adiabats'. Skew-Ts and tephigrams contain both moist and dry adibats, and the dry adibats are depicted by straight lines on a tephigram. This difference becomes more noticeable towards the top left hand side, where on a skew-T, the dry adiabats start to curve upwards.
If you're accustomed to tephigrams, a skew-T might appear at first to be very unstable – as if a thunderstorm were imminent. But that's only because the orientation of the temperature lines on a skew-T tend to curve away to the left more than with a tephigram.
And one final note when it comes to tephigrams. Entropy is usually denoted by the Greek symbol Φ, pronounced "phi". So instead of a T-P diagram, as with the skew-T, it's actually a T-phi-diagram. A 'Te-Phi-Gram'. Get it?