How are flow fields different from potential fields?

A potential field is a type of scalar field. In contrast, a flow field is a vector field. These are essentially just multidimensional arrays that are used as low level data structures for various pathfinding designs; there really is no "standard" algorithm equivalent to the ubiquity of A* for point-to-point pathfinding.

Typically, a potential field maps the "desirability" of a particular location on the terrain for AI entities (henceforth I will call them agents). They will try to travel in whichever direction will cause the greatest increase in potential. Hence, if you think of the potential as a height, they're playing a game of king of the hill. Once agents reach a local maximum, they stop moving. This can be a problem depending on how the field is generated, because agents can get stuck in corners and dead-ends.

One ingenious method of getting around this problem with potential fields is to allow the field to be modified in real-time. This is known as collaborative diffusion. In this system, "potential" acts somewhat like molasses or lava; it diffuses slowly toward any adjacent location with less potential. The target becomes a producer of potential, as if a mountain were constantly springing up beneath its feet. Agents become consumers of potential, simultaneously moving towards locations with greater potential, and reducing the amount of potential on their current location. In addition to solving the problem of local maxima, this means multiple agents will tend to move away from each other, taking different paths towards the target, and thus appearing to collaborate. Additionally, potential may naturally decay over time to prioritize the current and recent locations of the target over older locations.

For flow fields, I'll assume you're talking about a system similar to this one. In this system, you have two scalar fields and a vector field. You start out with a scalar travel cost field, which is integrated to find the total cost to travel from any starting location to the target location. Essentially, this integration field is a potential field, but instead of seeking higher potentials, agents seek lower potentials, and thanks to the integration that generates it, the potential field is well behaved: agents won't get stuck in a local minimum because there is only one minimum (the target location). Finally, a vector field is created by taking the gradient of the potential field (essentially the local derivative at each point in the field). This type of system can be thought of as an answer to the question "What direction would Dijkstra's algorithm (or A*, or any other algorithm that guarantees minimum path length) initially direct me to reach a specific target from every other location on the map?"

TL;DR A flow field is usually just the gradient of a potential field. If the number of agents in the simulation is significantly smaller than the number of locations in the field, then a potential field is probably better, because it uses less memory (a vector is larger than a scalar). On the other hand, if you have tons and tons of agents, you might be recalculating the same partial derivatives over and over for each agent, and caching them in a flow field might be beneficial.