The Deacon was an accidental result of the Engineers' Black Liquid that was originally used as a bio weapon as well as presumably creating life. The Liquid infected Charles Holloway, who impregnated Elizabeth Shaw with a Trilobite. That Trilobite eventually matured and impregnated an Engineer on LV-223, and the Deacon developed and burst from the Engineer's chest.
Differences from Xenomorphs
The biggest difference between Xenomorphs and Deacons is the host they come from. Just like all Xenomorphs, they gain different characteristics from their hosts. It is important to note how the birth of this creature is different from the majority of Xenomorph births, it was born with arms and legs, while most other Xenomorphs born of humanoid hosts lack legs and arms before maturing. Instead of having a second mouth like other Xenomorphs, the Deacon just had a second upper jaw that it could unhinge like a Goblin shark.
Unlike most Xenomorph Chestbursters encountered by humans, the Deacon used the sharp back end of its head to tear a way out of its host.
Behind the Scenes
There is apparently some confusion over whether or not the Deacon is intended to be some kind of Xenomorph, largely brought about by the major script revisions made on Prometheus during development (originally, the Xenomorphs were to feature in the film) and as a result of the production team occasionally referring to the creature as an Protomorph or Proto-Xenomorph (terms carried over from earlier script drafts).
However, evidence suggests the Deacon is not supposed to be a Xenomorph breed, or even a progenitor to the Xenomorph race. A mural of a fully-grown Xenomorph, resembling a Drone, can be seen on the wall of the Engineer Temple on LV-223, indicating that the Xenomorphs already exist by the time of Prometheus. The film's co-writer, Damon Lindelof, has stated, "I felt that the punchline of Prometheus was going to be that there is human DNA in what we have come to know as the human Xenomorph." However, his key use of the word "was" confirms this concept was eventually dropped from the final film.
Furthermore, the Deacon is not a creature created by a Facehugger impregnating an Engineer, but rather is created by a Trilobite. In the film's original script, an Engineer did become impregnated by a Facehugger, leading to an Engineer-spawned Xenomorph, although this was later changed when 20th Century Fox pressed Ridley Scott to remove the Xenomorph from the story altogether.
Despite not being a Xenomorph, the Deacon shares undeniable similarities with the former species, including the use of a living host as a means of gestation. Both the Deacon and the Xenomorph also possess a skeletal frame, an elongated head and an inner-jaw (although the Deacon's inner jaw is notably different to that of the Xenomorph). These similarities have likely only fueled opinions that the Xenomorph and the Deacon are supposed to be related.