I just wanted to fess up I am the only person who has voted "yes". Perhaps the distinction is too subtle for a fuzzy logician like myself. I perceive "a" and "not a" (and "b and "not b") as identical conceptual units, since every negative definition is also a positive one, and vice versa.
The difference is that each expression only deals with one set of states for A or B.
I'm going to re-write things a bit so they make more sense to me. This may cause them to make more or less sense to you:
The first expression could be written as "If A = FALSE, then B = FALSE".
The second expression could be written as "If A = TRUE, then B = TRUE".
The first expression only addresses what happens when A is false. It says nothing about what happens when A is true.
There are a few possibilities where the first expression does not imply the second:
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B is always false. It isn't dependent on A at all. An example of this would be if A is whether a light switch is on and B is whether the bulb is illuminated. If the light bulb is burned out, the first sentence is correct: with the switch off (i.e. A = FALSE), the bulb is dark (i.e. B = FALSE). In this case, though, the second sentence is not correct: with the switch on (A = TRUE), the burnt-out bulb will still be dark (B = FALSE)
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B depends on other variables. Based on our information, when A is true, B is undefined. An example of this would be boiling water: if you do not fill the kettle with water (A = FALSE), you cannot boil water (B = FALSE). However, if you do fill the kettle (A = TRUE), whether you can boil water (B) is dependent on whether you turn the stove on (C = TRUE) or not (C = FALSE). If we do not know whether the stove is on, we cannot say whether a kettle filled with water will boil. If B is dependent on other variables, even if we can say that A = FALSE implies B = FALSE, we cannot automatically infer from that that A = TRUE implies B = TRUE.