What is the difference between a ladies shaft and a seniors shaft on golf clubs?

Something labeled senior flex will probably be more flexible than regular. There is no real standardization among shaft makers, but typically they are ordered (from most to least flexible)

L - Ladies

A - Senior

R - Regular

S - Stiff

X - eXtra stiff

The rule of thumb is that slower swingers do better with more flexible shafts, because if timed right the flex can add some power and height to the shot.

The "A" designation between Ladies and Regular flex doesn't stand for anything in particular. They couldn't use "S" for the Senior flex because it was already being used as the designation for "Stiff" flex. The abbreviation for Senior, "Sr.", could also be confused with a "Regular/Stiff" flex. So in an attempt to avoid confusion (or possibly add a little), the letter "A" was decided upon as the designation for the Senior flex shaft.