I did a google search, a lot of math which admittedly I have limited skill with. In layman's terms, what is the difference between the two and how do those differences affect the final product?
A traditional horn has a driver, an expansion chamber, a throat and flare, which can be exponential, Tractrix, or parabolic. It belongs to the family of horns, Like the French Horn, Tuba etc.
Here is a horn speaker.
The low frequency cut off is set by the diameter of the mouth. However when placed in a corner, this does add to the flare and extend bass response. However for a good sub a traditional horn would have to be huge.
There is also the tapped horn popularized by Danley, that is tapped because the driver crosses two compartments of the line. This allows for deeper bass in a smaller horn.
A transmission line also comes in various guises. The traditional lines are specialized Gaddeckt organ pipes. Therefore is a stopped pipe open at one end and not both, like an open pipe.
A stopped pipe has a fundamental frequency and odd harmonics. To lessen harmonic output, the driver is placed a third to quarter the length of the pipe at the closed end.
At the closed end of the pipe the pressure changes are high, and control cone movement well. At the lower end there is a null of pressure, but air flow is maximal.
The pipe is tapered to broaden the resonance, usually with the widest part at the closed and and the narrowest part at the open end. This is the opposite of horn expansion.
The pipe is designed to make Fp match the driver Fs, like you do with a tuned box, allthough this relates to length and not volume. The volume of air in the pipe is largely determined by driver VAS.
A line without damping has an impedance curve like a ported box. However the pipe is then damped with stuffing just, and only just to the point of suppression of one peak of impedance. The line is then critically damped and reproduction will be non resonant, with useful bass augmentation and good control of driver cone displacement. At the point of critical damping the line will roll off second order, 12 db per octave, rather then the fourth order 24 db per octave roll off, of an undamped line.
Here is my center channel TL under construction.
And then it is damped.
Unfortunately some authors have wrongly referred to pipe bases speakers as horns, which is not correct. This causes enormous confusion.