Dictionaries give both dysunction and malfunction the meaning 'failure to function properly'. Are they complete synonyms?
They are not exact synonyms.
Dysfunction is defined as:
any malfunctioning part or element: the dysfunctions of the country's economy.
The adjective of dysfunction is dysfunctional.
1. failure to function properly: a malfunction of the liver; the malfunction of a rocket.
verb (used without object)
2. to fail to function properly.
The corresponding adjective is malfunctioning.
Per the definitions above, dysfunction (noun) is a malfunctioning constituent, whereas a malfunction (noun) is the failure to function. So you would say the dysfunctions in the launching apparatus caused the malfunction of the launch.
However, dysfunction has another meaning, which I believe is worth a dictionary entry: as Kit notes, general usage shows that dysfunction/dysfunctional usually applies to an ongoing abnormality of the object; it might still work, but it doesn't work as intended. Malfunction, on the other hand, connotes a breakdown of sorts, after which the object ceases to function.
Another difference is that malfunction can be a verb, and dysfunction cannot. This makes sense when considering the following analogy: failure:fail::failing constituent/abnormality:[what?], corresponding to malfunction [noun]:malfunction [verb]::dysfunction:[nothing].
It is true that both malfunction and dysfunction have similar meanings, but the prefix mal- means (approximately) bad while the prefix dys- means (approximately) abnormal.
So although drɱ65 δ's links above show that the definitions for malfunction and dysfunction are synonymous, I argue that (despite what Dictionary.com says) dysfunction and malfunction are generally used for different things.
For instance, it sounds exceedingly wrong to me to hear "a malfunction of the liver." It should be "a liver dysfunction." Also, we talk about dysfunctional relationships or families, not malfunctioning ones. So dysfunction is generally applied to organic things, or abstract entities or processes. A dysfunction does not stop something from working, but it does mean that the thing is not working properly. This is a chronic case, happening over a less defined period of time.
"It was caused by a computer dysfunction" also sounds wrong. Usually, we'd say "it was a computer malfunction." As an additional example, wardrobes sometimes malfunction. Malfunction is used for things that are operated on, like mechanical devices. A malfunction is much more like a break; something stops working or fails to work. It is an acute case, happening in a short, defined time span.