Why didn't my Deathtouch creature kill my opponent's Protean Hydra?
Protean Hydra has two abilities that say

If damage would be dealt to Protean Hydra, prevent that damage and remove that many +1/+1 counters from it.
Whenever a +1/+1 counter is removed from Protean Hydra, put two +1/+1 counters on it at the beginning of the next end step.

And the rule for how Deathtouch damage works is 702.2b:

A creature with toughness greater than 0 that’s been dealt damage by a source with deathtouch since the last time state-based actions were checked is destroyed as a state-based action

So, here's the sequence of events in your scenario:

We get to the combat damage step, and the Spider is assigned to deal 2 damage to the Hydra.

The Hydra's ability prevents that damage, and instead 2 +1/+1 counters are removed, making it a 10/10. Since the damage was prevented, the Hydra wasn't dealt damage by a source with Deathtouch, so it doesn't die.

At the beginning of the end step, because of the Hydra's other ability, 2 +1/+1 counters are put on it for each one removed earlier, making it a 14/14.


You said that the Hydra ended up as a 13/13 instead. That could happen if the Protean Hydra was actually a 11/11 originally, or if the Spider actually only had 1 power, or it could simply be a bug in the game.