I may be wrong but i think that a food chain shows where energy goes from one thing to another but a food web shows how multiple food chains within any given area interacts.
Like David says, a good chain show the direction of energy flow through a single path of a food-web. For example:
Fox
^
Rabbit
^
Grass
Whereas a food-web has many other connections. In the same example, energy from grass would flow not just to rabbits but also geese, sheep, goats, bison, caterpillars, and from the energy would flow to many different predators.
So really the key difference is how complex they are.
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Sam R commented on :
Like David says, a good chain show the direction of energy flow through a single path of a food-web. For example:
Fox
^
Rabbit
^
Grass
Whereas a food-web has many other connections. In the same example, energy from grass would flow not just to rabbits but also geese, sheep, goats, bison, caterpillars, and from the energy would flow to many different predators.
So really the key difference is how complex they are.