There can be a difference but it is easier to see with other examples.
Let’s take hobbies.
Er spart an seinen Hobbies — He wants to spend less on his hobbies.
Er spart bei seinen Hobbies — He saves money during his hobbies (perhaps his hobby is a really cheap one so he spends less during the time he does this than he would normally.)
Mostly c.p. had it right with his/her first sentence.
One uses accusative to denote the thing one saves and bei for the process
Er spart am Benzin oder er spart beim Autofahren indem er ökonomischer fährt.
He saves on (?) gas or he saves while driving by doing it more economically.
In both cases he pays less money for gas. In the first part he just buys less, in the other he needs ness because he’s more careful how much he uses.
Er spart am Essen oder beim Essen ist — is not as easy to understand as Essen can be both, the food and the process of eating. And this is where the difference is (or at least can be).