What is the difference between the cost estimate and the bill of quantities?

In nutshell, Cost estimate includes bill of quantities .

A BOQ is an estimation prepared in concurrence with the approved drawing estimating the material quantities to be consumed in construction .It may or may not include costs to each n every quantity of items.

In Cost estimates ,quantities are assigned with costs of each item giving an overall picture reflecting the funds required for execution .Cost estimates are useful for finance dept to plan for funds or to quote for bidding process. BOQ are useful to Site engineers to plan for human resources , materials ,deadlines ,execution etc.