  The connection between model-completeness and elimination of quantifiers is as follows.  Robinson showed that a theory is model-complete if and only if it has elimination of quantifiers down to existential formulas (i.e. formulas that either are quantifier-free or consist of one or more existential quantifiers followed by a quantifier-free formula).  So theories that have quantifier elimination are model-complete, but the converse need not hold. Still, showing that a theory is model-complete is sometimes a useful first step towards showing that it has quantifier elimination.