Completenes means that every basket of goods in some set previously defined is comparable with the use of a complete preference. Now, with the additional assumption that the preferences are transitive, we can prove that this preference can be represented by an utility function. That sums up the importance of the axiom.
Normally continuity is defined as an additional assumption for utility functions in text-books of microeconomics, but why completeness does not imply continuity, or the latter, continuity, does not imply completeness?
Assumptions for existence of a utility function
In your question you imply that the only assumptions needed on preference to produce a a real-valued function (a utility function) that represents those preferences are completeness and transitivity. This is incorrect. To represent preferences with a real-value function, you need (1) completeness, (2) transitivity, (3) continuous preferences, and (4) local non-satiation. You can find a proof of this fact in Microeconomic Theory by Mas-Collel, Whinston, and Greene. A simpler proof using strict monotonicity in place of local non-satiation can be found in other books (Reny and Jehle).
The assumption of continuous preferences
To clarify, the axiom of continuity is the assumption that the preferences are continuous, not the resulting utility function. The preference relation $\preceq$ is continuous if, for any bundle $x \in X$, the set of all bundles at least as good as $x$, $$ \succeq(x) \equiv \{y \in X \mid y \succeq x \}, $$ is closed. In contrast to a continuous utility function, note that a set of preferences has infinitely many possible representations. For example, the preference relation on levels of money $m \in \mathcal R$ such that $m_1 \succeq m_2$ iff $m_1 \geq m_2$ is continuous as describe above. Now, notice that these two utility functions are both valid representations: $$ u_1(m) = m $$ and $$ u_2(m) = m + \boldsymbol 1\{m \geq 2\}. $$ These are both valid because $u_i(m_1) \geq u_i(m_2)$ iff $m_1 \geq m_2$, $i = 1,2$. Thus $u_i(m_1) \geq u_i(m_2)$ iff $m_1 \succeq m_2$, $i = 1,2$.
Example of Preferences that are not continuous
The classic example of preferences that cannot be represented with a real-valued function are lexicographic preferences. The idea behind the proof is that a utility function can represent the ordering along the first category in the lexicographic ordering, but afterwards there "are not enough numbers left" to represent the others. See Mas-Collel p. 46 for details.