Is a deductive fallacy a failed enthymeme or syllogism?

It could be both; these terms are not mutually exclusive.

An enthymeme is a syllogism with an implied line. A syllogism is an argument format that may be deductive or inductive. Therefore, if a deductive argument is fallacious, it is possible that the fallacy arises out of a failed enthymeme; that is, too much liberty has been taken with the implied portion of the argument. It is also possible that a syllogism has been constructed fallaciously. Here are some examples to illustrate:

Enthymeme:

Spot is a mammal because he's a dog.

The part that is implied here is that all dogs are mammals. This can be left out because it is generally understood that dogs are mammals. This is a successful enthymeme.

Failed enthymeme:

Abortion is wrong because it is murder.

The part that is implied here is that a fetus is a person, because only persons can be murdered. In this case, a significant portion of the argument is being glossed over, and it is the part that most people who disagree with the argument are going to be concerned with. Because we cannot assume that it is generally accepted that a fetus is a person, this enthymeme fails. This premise, though suppressed, is false.

Syllogism:

If I finish my homework, then I will go to the movies.

I finished my homework.

Therefore, I will go to the movies.

This is a successful syllogism, because the conclusion follows from the premises.

Here is a syllogism that results in a deductive fallacy:

If you are a Tea Party member, then you are racist.

You are a Tea Party member.

Therefore, you are racist.

In this case, the conclusion follows logically from the premises, but you can see how it still might not be true. Because it is possible that a single Tea Party member isn't racist, you know the argument is fallacious. The problem arises from the first line, which states that all Tea Party members are racists. There is no reason to believe this is true. Because the syllogism creates a valid argument but still isn't true, it contains a deductive fallacy.

it really depends on the deductive fallacy that has failed. Is it the fallacy of suppressed evidence? Likely an enthymeme.