The key difference between the two is their cutting action.
A bypass pruner features two blades that slice past each other, much like scissors, making it ideal for pruning live wood. An anvil pruner has a single straight blade that closes onto a flat edge (the anvil). This action tends to crush the stem as it cuts, which is why anvil pruners are best reserved for dead or brittle wood. For pruning living plants, a bypass mechanism is the superior choice.