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<blockquote>Does Pr (probability) of winning jackpot differ between
1. picking 20 different #s, for the same draw
vs.
2. picking 10 different #s, for 2 different draws ?</blockquote>

Yes.  Consider the limiting case where there are only 20 possible choices.  Playing all 20 numbers guarantees a win.  Playing 10 out of 20 choices results in probability of winning of &frac12;.  Therefore playing 10 choices each in two different draws results in a probability of winning of &frac34;.

Note here that "winning" is used only to mean picking the winning number.  It does not imply being better off at the end or a positive expected value.  In any remotely competent lottery, it would cost more to play every possible number than you get by winning.