Both of the above are pretty fair one sentence answers and relatively accurate. I wrote the rest below and then got the "Others have posted" message. Maybe a little more detail will helo. If not... Oh Well.
Those RRRRs said:
Is the HD AM Band affected by lightning like the terrestrial AM Band? And how is it that we can put a man on the moon but can't seem to "fix" the terrestrial AM Band so that the lightning problem no longer affects it? Or is it that no one has bothered to fix it?
I'm sure there ore others with a better explanation, but I'll start off and get you confused.

The difference is how the radio wave is changed to produce the sound.
You start with a radio wave on a certain frequency or wavelength and go from there. If you receive this wave with your radio... you get silence. Not "static" as in no station, but silence.
There are three other things you need to know to get how this works...
1) AM or AMPLITUDE Modulation, changes the amplitude or SIZE of the radio wave depending on what is being transmitted. Changes in the SIZE of the wave are detected by your good old AM radio and turned into sound. The louder the sound, the bigger the Amplitude or Size of the wave.
2) FM or FREQUENCY Modulation changes the Frequency of the radio wave depending on what is being transmitted. Small changes in the frequency of the wave are detected by your FM radio and turned into sound.
3) Lightning is (for lack of a better term) a giant electrical ZAP! It make a big amplitude wave across almost all of the radio bands.
So when AM is transmitting and FM is transmitting and there is a lightning CRACK... here's what happens...
A really Big "High Amplitude" wave is created by the lightning. It is on the AM band and on the FM band. Here's how your Kitchen radio on AM and your Office Radio on FM react to that big wave...
The Kitchen radio on AM reacts to the Really Big wave by making a really big sound. A Staticy sound, because unlike a song or voice, it's just really a really loud "Electrical" noise. As you know if you get close to the lightning strike it also makes a really loud SOUND as well.
The Office radio is listening to Frequency changes on FM. While it "can" become confused, it is looking for the Frequency changes in the radio wave it is listening to. If the Lightning put a really big wave over what the FM radio is listening to, it can overload your FM receiver and cause momentary confusion, but when an FM radio is confused, it tends to not see a frequency change and basically creates nothing Including static. If the lightning is furthur away, and the wave is not as big, it might not affect your FM radio at all.
As I understand it, the advantage (Such as it is) for HD is that it sends the data more than once (Which causes the delay). Since it sends it more than once, it has more than one chance to get it right. Therefore it is more RESISTANT to lightning. I would suspect AMHD is still affected by moderate storms (but better than analog) and FMHD is still affected by SEVERE storms (Tho better than analog).
It is ironic actually. We can put man on the moon, but I suspect we will NEVER conquer the static of mother nature.
I'll now gladly step aside and let the carreer engineers correct the finer points... (Or show me to be a complete numbskull.)
Clouseau