Continue is one of the different meanings of Proceed. Proceed (v) 1. begin a course of action. "the consortium could proceed with the plan" synonyms: begin, make a start, get going, move, set something in motion; *do something after something else. "opposite the front door was a staircase which I proceeded to climb" *(of an action) carry on or continue. "my studies are proceeding well" >LAW *start a lawsuit against someone. "he may still be able to proceed against the contractor under negligence rules" synonyms: take to court, start proceedings against, take proceedings against, begin an action against, start an action against, sue "there is not enough evidence to proceed against him" 2. move forward. "from the High Street, proceed over Magdalen Bridge" synonyms: go, make one's way, advance, move, move forward, move along, progress, carry on, press on, push on "he turned off the road and proceeded down the long drive" antonyms: stop *advance to a higher rank, status, or education. "he did not proceed to university in his seventeenth year" 3. originate from. "his claim that all power proceeded from God" synonyms: originate, have its origin, spring, stem, come, derive, arise, issue, flow, emanate, descend, result, follow, ensue, begin, emerge, start "his claim that all power proceeded from God"

In many/most situations, to proceed and to continue mean exactly the same thing. Where there are differences, those differences in usage are rather subtle. If you Google "proceed or continue" you will find a number of in-depth explanations.