A Subplot is a personal choice, you can select one from the available Subplots in the story already, or create your own.
A Goal is one that is assigned by the Narrator, either directly to you, or available to be picked up.
All Subplots start with 5 cards, whereas a Goal can start with any number of cards as dictated by the Narrator.
In this way, the Narrator can give incentives for players to do certain things, whereas the Subplot is personal incentive for your own character to accomplish his own goals.
Also, here is a quick explanation of the cards from the creator of Storium found on the Kickstarter Comments:
The idea of Goals and Subplots is that they are rewards for playing
your character in a certain way over an extended period of time. Goals
are ways for the narrator to encourage certain long-term plot
directions without forcing it on players, whereas Subplots are
long-term plot directions that players can choose for themselves. In
both cases, the player gets a reward for having committed to that
long-term direction and thus helped to advance the story. These cards
are limited to one play each per scene because otherwise a player
could "spam" their cards, earning the reward too quickly and
undermining the entire storytelling purpose of the feature.