The difference between automatic stabilizers and discretionary fiscal policy can be seen in their names. The former is something that happens automatically. The latter, by contrast, only happens at the discretion of the people who make policy in the federal government. Let us explore what this means.
Automatic stabilizers act to stabilize the economy without any action being taken by any policy makers. Automatic stabilizers are already written into law and therefore act without any new impetus from Congress. For example, unemployment insurance payments are automatic stabilizers. Congress has already written laws that dictate who will get such payments under what conditions. When those conditions arise, the payments are made automatically, without Congress having to take any further action. As the payments are made, they help to stabilize the economy by putting money in the pockets of unemployed people. This helps to prevent aggregate demand from decreasing too rapidly during a recession.
By contrast, discretionary fiscal policies must be enacted by Congress. They do not happen automatically. An example of this would be public works projects that are meant to increase aggregate demand by putting people to work. When a recession happens, these projects do not simply start on their own. Instead, Congress has to approve them. Congress does not have to do so, thus making them discretionary.
So, the difference between these two is that discretionary fiscal policy must be enacted by Congress in response to problems while automatic stabilizers work without Congress having to take any new action.