The difference is between a to-infinitival as a relative clause, and a gerund-participial complement to preposition to.
As a relative clause, the key could be subject and take agent role in the situation:
the key to choose wisely = the key which chooses wisely
?the key chooses wisely
It would seem a bit odd to put an inanimate object in the agent role, but it does work sometimes:
Education is the key to get them out of poverty.
Education gets them out of poverty
the key could also be understood as object, with the subject left understood
the key (for you) to choose ___ wisely
You choose the key wisely
This seems a bit awkward with key, but does work just fine with other words:
the decision (for you) to make now is whether to give them the money
you make the decision
Contrast this with:
Education is the key to (us) getting them out of poverty.
We get them out of poverty.
Here, with to separating key from getting it is clear that key is neither subject nor object in the clause headed by getting, but rather the means by which this is done.
The clear choice for the advertisement is this last construction as the key is neither doing the choosing nor the thing being chosen, but the means by which one may choose wisely.