HP RPN is stack based, which means you can write functions that operate on the last n entries on the stack (last n results), without monkeying around with up arrow to select prior results to insert them into an expression you're building, and without assigning every intermediate result to a variable.
<boring calculation> right side: 2.5536
<boring calculation> right side: 3.5*e
<boring calc> right side: 34*Pi
You want to do some calculations with those three values. Let's say (3.5e)^2.5536/(34*Pi).
[ROTdown] [swap] [x^y] [swap] [/] (typically rotate and swap are mapped to softkeys)
probably the most efficient way to do it on a stock TI is
<boring calc1> -> sto X
<boring calc2> -> sto Y
<boring calc3> -> sto Z
That works, except then you have to store all intermediate results and good luck keeping track of variable names if you have more than a few. If you don't save as variables, you're in up arrow hell selecting each prior result: [up arrow * 3, enter] ^ [up arrow * 5, enter] / [up arrow, enter] [enter]. And it gets worse as calculations get more complicated.
tl;dr: trying to manipulate a bunch of different past results on a TI, without having them stored as variables already, is a PITA. On a stack-based rpn calculator like the HP48/49/50, it's a breeze.
Disclosure: I started out in algebra 1 with a TI 81, then an 83, then switched to an HP48gx and never went back. My HP48 died a while back (screen damage) and I wasn't using a calculator anymore so I didn't replace it, but I'm about to get a new one... probably an HP50g unless the RPN hacks for the TI84+se/TI89ti are any good. Or I might just rely on the hp48 emulator and ti 82-86 emulators for android.
My experience with calculators in school was that they're of far more use for science/engineering than for pure math classes. There's very little in real math you need a calculator for; any examples or tests could be designed to have numbers that are easy to work out by hand. Therefore, in math classes, either they're artificially designed to teach calculator skills (in which case the class probably targets TI skills and you need a TI), or you're not getting nearly as much use out of a calculator as science/engineering students would, who are constantly using a calculator or computer in at least half of their classes. In an ideal world, I think math classes should be using computers where necessary, rather than using calculators. Would you rather aid learning of functions, graphs and matrices on a slow device with a tiny grayscale screen, or using Mathematica/Octave/R/etc.?
I don't know where HP went wrong in failing to win educational mindshare. Maybe it's because they weren't really trying, while TI put a lot of effort in the 90's into marketing and classroom usability -- for instance, my geometry teacher beta tested the TI92 for geometry the year before I took the class. TI's (gimmicky) lab data collection interface, I suppose, got science teachers hot and bothered. HP probably knew that professional engineers and financial professionals liked their HP rpn calcs the way they were, and so they saw no reason to compete with TI in the classroom.
HP's built-in (I think it was built in?) units library, attaching units to values, and automatic dimensional analysis during computation made doing scientific computations *fun*. And before I get the typical "learn how to do it yourself!" flames, I was perfectly capable of doing it myself; the point is that with your calculator keeping track of units, you can see every step of the way whether you might have gone wrong, because the units will start looking wrong if you know what you're trying to do.
So here we are in 2011, TI calculators don't have built-in RPN, the TI 8x series don't have built-in dimensional analysis, and they're weaker in a few other mostly science/engineering areas as well. Why? Because TI's been too busy locking down their firmwares and spending too much effort trying to turn calculators into computers. If I had no computer and I wanted to do 3d graphing, an nspire would be a no-brainer. Otherwise, though? I don't see the point.