BASICally, it's like C, but for a calc.

It depends on what you're trying to do.  Somethings take almost the same amount of time like "Disp", "Output", and "ClrDraw" for instance because they are basically the same routines.  The only speed increase there is the absence of the interpreter.

Other things like storing to variables, math, and reading from lists are hundreds of times faster than BASIC becasue you are dealing with 16-bit integers instead of 72-bit floats.  Native assembly is even faster still because you can use registers for temporary storage and decide when you need the faster 8-bit operations instead of the 16-bit ones, but the difference is only 1.5 to 2 times the speed generally.

The rest of the commands like sprite drawing, grayscale, copying data, etc. are pretty much the exact same speed as regular assembly because its literally replacing those with the same routines you would use if you were programming in regular assembly except I've spent the time to optimize them to be as fast as possible.

I can't really compare Axe to BASIC much more than the math commands becasue things like sprite drawing don't exist in BASIC.  Pixel drawing is the same speed in BASIC and in Axe if you're updating the screen after every pixel.  But if you draw all the pixels first and then update the screen when you're done, you can fill the entire screen in about half a second since updating the LCD is so slow.  The best way to see "how fast it is" is to just try running some programs like "How long does it take to draw 10,000 sprites?"  (about 3 seconds)  "How long does it take to do 10,000 multiplications?" (about 2 seconds) etc.

That includes aligning and clipping in every direction which sometimes is not used in regular assembly when you don't need those and I have a feeling a lot of that time actually went into the for loop I was using since it took almost a second even with nothing in the loop.