The PMBoK sometimes uses them interchangeably: "An objective is defined as an outcome toward which work is to be directed, a strategic position to be attained, a purpose to be achieved..."
Objective can have other meanings however, such as an objective decision is based on fact, while a subjective one is based on opinion. Objectives are also used in the context of requirements. While a requirement is something that must be achieved by a project, and objective is something desirable, but not absolutely necessary. "The product must support 4 users" is a requirement. "The end users should require minimal training." is an objective. Saving Changes...
Kiron explained it very nicely. The objective is what you're trying to accomplish, and the purpose states why you need the objective.
A project statement might read:
We will [OBJECTIVE] so that [PURPOSE].
It's important to state the "why" so the project team can stay focused on the objectives and adapt if necessary. Throughout the course of the project, they might discover a better way to accomplish the objective, or market conditions might change and the objective becomes obsolete.
Keith is also correct, though, that the two terms are often used interchangeably. An organization might state the purpose as the objective. Saving Changes...