The three thermodynamic diagrams in common use in operational meteorology
The three plots below represent the main thermodynamic diagrams in use around the world.
SKEWT/LOG(P) 
STÜVE 
TEPHIGRAM 
they are intended to demonstrate the differences in appearance between the three versions, rather than be a strictly accurate representation of that particular diagram. For this reason, I have deliberately left off the labels!
This table will attempt to explain the differences between each diagram, and also explain a little about each printed 'line' on the charts, with the units usually used for the variable displayed.
Line on diagram 
units used (usually) 
SKEWT/LOG(P) 
STÜVE 
TEPHIGRAM 
degC 
straight & parallel; angled 45deg/slope to right. Equal spacing (linear); angled circa 90deg to Dry Adiabats up to 300hPa. 
straight, parallel & vertical. Equal spacing (i.e. linear). 
straight & parallel; angled 45deg/slope to right. Equal spacing (linear); angled exactly 90deg to Dry Adiabats whole diagram. 

hPa (or millibars in old money) 
straight, parallel & horizontal; increased spacing per unit pressure change with altitude. 
straight, parallel & horizontal; increased spacing per unit pressure change with altitude. 
very slightly curved upwards (i.e. convex towards top of diagram). [ not really noticeable for routine use.]; quasihorizontal; increased spacing per unit pressure change with altitude. 

DRY ADIABATS (lines of constant potential temperature for a dry air sample [ i.e. an unsaturated air parcel path. ] ) 
degC (but strictly, and sometimes found, degrees Kelvin are used) 
curved: approx. 45deg to left near 1000 hPa, decreasing to within 10deg of vertical near 100 hPa. 
straight  sharply angled to left  gently convergent to left. (meet at a theoretical point where P=0; T(K)=0) 
straight & parallel; angled 45deg/slope to left; Equal spacing (linear); angled exactly 90deg to Isotherms whole diagram. 
SATURATED ADIABATS (lines of equivalent potential temperature for a saturated [or 'wet'] air parcel path.) 
degC (but strictly, and sometimes found, degrees Kelvin are used) 
curved  but not constant; on righthand side of diagram, curve starts right and bears left above 400 hPa; on lefthand side, curve starts left and quickly become parallel with Dry Adiabats. 
slightly curved to left with height  curve minimal lefthand side of diagram; a gently increasing left turn on right hand side. 
The only notably curved lines on this diagram: On the righthand side, starts slightly right before curving left; on lefthand end, curve all to left. On most diagrams, not shown above about 50degC. 
SATURATED HUMIDITY MIXING RATIO (lines of constant saturation mixing ratio with respect to a plane water surface.) 
g/kg ( i.e. ratio of mass of water vapour in given volume to the mass of the dry air in that sample. ) 
quasistraight*; angled to right, at less than 45deg to the vertical; gently convergent to a point well above the top of the diagram. (* for practical work can be regarded as straight & parallel) 
quasistraight*; angled to left, at less than 20deg to the vertical; gently convergent to a point well above the top of the diagram. (* for practical work can be regarded as straight & parallel) 
quasistraight*; angled to right at less than 45deg to vertical, i.e. less slope than isotherm. (* for practical work can be regarded as straight & parallel) 
… and diagrammatically, I have attempted to highlight the various lines here…. the diagrams are not true representations of each style or necessarily to scale, but are close enough to pick out the major differences listed above:
ISOTHERMS:
SkewT/LogP 
Stüve 
Tephigram 
ISOBARS:
SkewT/LogP 
Stüve 
Tephigram 
DRY ADIABATS:
SkewT/LogP 
Stüve 
Tephigram 
SATURATED ADIABATS:
SkewT/LogP 
Stüve 
Tephigram 
SATURATED MIXING RATIO ISOPLETHS:
SkewT/LogP 
Stüve 
Tephigram 
Some definitions/abbreviations: (not already given elsewhere by hyperlink)
degC 
degrees Celsius 
describes the 'parcel path' on a thermodynamic diagram when that parcel is unsaturated: i.e. 'dry'. Taken to be 3degC per 1000 ft, or 9.8 degC per km. 

the ratio of the amount of heat absorbed by an object in undergoing a reversible thermodynamic process to the absolute temperature of the object (dQ/T) is defined to be the increase in entropy. 

g/kg 
grams (of water vapour) per kilogram (of dry air) 
hPa 
hecto Pascal … same as millibars. 
is defined as the temperature an air parcel would have, if it were moved vertically (upwards >> decreasing pressure >> expansion; downwards >> increasing pressure >> compression ), from its existing pressure and temperature to a standard level (usually defined to be 1000 hPa). Provided the parcel remains 'dry' or more strictly unsaturated, then the rate of cooling (upward motion), or warming (downward motion) occurs at the Dry Adiabatic Lapse Rate (DALR), which is 9.8 degC per km ( or 3 degC per 1000 feet ). This value is constant. Once a parcel becomes saturated (i.e., the initially unsaturated parcel is lifted until it cools to its dewpoint temperature), then the subsequent release of latent heat of vaporization offsets the cooling rate, and the Saturated Adiabatic Lapse Rate (SALR) is consequently less than the DALR. (see Saturated Adiabats) 

describes the 'parcel path' on a thermodynamic diagram when that parcel is saturated: Taken to be, very roughly, 1.5degC per 1000 ft, or 5 degC per km IN THE LOWEST FEW HUNDREDS OF MILLIBARS OF THE TROPOSPHERE. Towards a temperature of minus 50degC, tends to the Dry Adiabatic Lapse Rate, as air at lower temperatures holds less and less moisture, hence less offset from the latent heat release. 
Advantages and disadvantages
Very much in the eye of the beholder here. The tephigram is regarded as nearperfect for strict thermodynamic calculations, and its large angle between isotherms and dry adiabats renders it the most effective as assessing degrees of stability. However, the skewT also has this property, and tephigrams aren't exactly plentiful on the net. The skewT though has curved dry adiabats, though for short vertical calculation perhaps not a major problem. Indeed, for a parcel undergoing saturation, you are going to have to cross over to the saturated adiabats anyway, which on all the diagrams are curved. The Stüve is clean and simple in that it has the most straight lines, and is perhaps intuitive in that isotherms are vertical against a pressure plot. The disadvantage is that the angle between isotherm and dry adiabat is not as great as the other two versions. Again, constant use would offset this minor difficulty. For those used to the tephigram, the skewT is most like what you are used to.