Here's how it works. In the game state with no one on base and no one out (hereafter abbreviated 0-000), you have a probability of scoring exactly zero runs from then on (P0), a probability of scoring exactly one run from then on (P1), and the probability of scoring more than one run from then on (P2+ = 1 - P0 - P1). Conversely, the probability of no multi-run inning is P0 + P1 = 1 - P2+.
After a lead off home run, you return to the 0-000 game state. Only now, in order for there to be a multi-run inning, you only need to score one more run. Therefore, the probability of no multi-run inning is now just P0. The probability of a multi-run inning has become P1 + P2+.
After a lead off walk, you enter the game state 0-100 (man on first, no one out). You still need to score two more runs. As before, you have a probability of scoring zero runs P0', one run P1', and two or more runs P2+'.
In order for the lead off walk to be more valuable, P0 would have to exceed P0' + P1'.
Now, P0 is roughly 0.72. P0' is roughly 0.58. P1' is roughly 0.25. Therefore, P0' + P1' is equal to roughly 0.83. Therefore, the probability of scoring one run or more from state 0-000 is 0.28. The probability of scoring two or more runs from state 0-100 is 0.17. (All of these numbers are based on Keith Woolner's "An Analytical Framework for Win Expectancy" from Baseball Prospectus 2005.)
The difference is roughly 11%. Most (all?) of this will be accounted for by the fact that the man on first can be doubled off and the man who hit the lead off home run can't.
So that's the math. But do you really need it? A home run is one guaranteed whole run that no one can take away. The lead off walk increases your odds of scoring exactly one run by roughly 14%. The lead off home run increases those odds by ONE HUNDRED FREAKING PERCENT.
**EDIT** As pointed out in comment number one, referencing the probability of scoring one run is misleading, as scoring one run and scoring zero runs both count for nothing for the purposes of counting multi-run innings. The ninth-inning analogy is apt: it's the second runner scoring that is important and neither the lead off walk nor the lead off home run will have a great affect on what that second runner does. What is important is that the lead off home run eliminates the probability of the double play and the lead off walk does not. In other words, it is the out that is important, not the run. Point well taken. **END EDIT**
Tim, do us a favor and stop bringing this up as if it's surprising. You will sound a whole lot more intelligent and