Yeah, the top of page 10 is clearer about the "use it or lose it" part:
The cards usually offer a power. the power of a card may only be used immediately upon fulfilling that contract (erecting that building). If it is not used at that time, the power is lost and may not be used at a later time in the game.
A small timing note: you make the choice to use the power, and how, before you reveal the replacement (for neutral contracts).
Distillery is a bit odd; my current interpretation is that you do the action (which might be fulfilling another contract, and its power) before your reveal the next card.