The extent of the change is seen in Stargate SG-1 1x14, "Hathor", when she turns Jack O'Neill into a new Jaffa.
So yes, the Jaffa are modified humans, and as seen near the end of the episode,
the loss of an immune system is part of the genetic makeup of the Jaffa. However, as we know thanks to Rya'c (Teal'c's son), this isn't from birth. The loss of the immune system appears to happen during puberty. (SG-1 1x12, "Bloodlines"). As of SG-1 7x10, "Birthright", requiring a symbiote at puberty is made explicit.
The reason for the existence of Jaffa is first mentioned in that same episode, and clarified in the Tritonin episode 6x10, "Cure":
It is so that the Goa'uld symbiotes can better adapt to their host body. Without a Jaffa to adjust the symbiote to human physiology, a blending only has a 50/50 chance of succeeding. And if it fails, both symbiote and host die.
Quick addition, also from SG-1 7x10:
Carter: "Teal'c, these children have been genetically altered to depend
on symbiotes, ..."