What is the difference between preferences lacking "completeness" and being indifferent? If two preferences are complete, at least one must have a relationship to the other.  If neither has a relationship, doesn't that mean that the consumer doesn't care which one he/she purchases?
As alluded to in the comments the distinction is roughly:


Indifference: The decision maker knows she will receive the same utility from the consumption of $x$ or $y$.
Incompleteness: The DM does not know her preference between $x$ and $y$. (Note, this could stem from a lack of information, or because no preference exists)


So, from a conceptual vantage, the difference is that indifference is the existence of knowledge, whereas incompleteness is a lack of knowledge. While interesting in a foundational/philosophical sense, the distinction is subtle and likely not the reason economists have spent so much thought on the subject. However, the difference has behavioral implications! and this is important for economics.

For example, imagine various choices between "pizza", "hot dog", or "hot dog and $\$0.05$". Perhaps, the decision maker has never had pizza or hot dogs and therefore finds them incomparable. As such, from the choice set $\{p,h\}$ she would choose either (since neither option is known to be inferior[1]). So $C(\{p,h\}) = \{p,h\}$.  Now, the decision maker also likes money, so although she does not know her preference for a hot dog, she will always choose more money to less, all else equal: $C(\{h, h+5\}) = \{h+5\}$. But her indecisiveness regarding the two foods might not be resolvable for 5 cents, so, it is reasonable that $C(\{p, h, h+5\}) = \{p, h+5\}$. But this violates the weak axiom of revealed preference. Such a choice function cannot be rationalized by a complete and transitive relation.

Notice if the decision maker was actually indifference between $h$ and $p$ then, $C(\{p, h, h+5\}) = \{h+5\}$. In summary: incompleteness can cause "thick" indifference curves, and this can violate the basic tenets of rational choice theory.



[1] There are other interpretations of choice under incompleteness, for example, allowing the choice function to return an empty set.