Plan vs mission - what is the difference in meaning?

Mission - something you have been sent to do (including by yourself). But you may have no idea how to carry out your mission.
Plan - the things you have worked out in order to achieve your mission - or anything else.

My mission is to get pizza. My plan is to drive to the local pizza house and purchase a takeaway. I could have a different plan for the same mission: phone for pizza delivery. Or even to cook my own pizza.

Mission = what you want to accomplish
Plan = the steps you will take to accomplish your mission

Mission

something that you feel you must do because it is your duty

the purpose or the most important aim of an organization:

http://www.ldoceonline.com/dictionary/mission

Plan

something you have decided to do:

a set of actions for achieving something in the future, especially a set of actions that has been considered carefully and in detail:

http://www.ldoceonline.com/dictionary/plan_1

Mission answers the question, "Why are we here?" Plan answers the question "What are we doing?" Mission is the expression of your core values and goals. A plan is a set of action steps to achieve some goal.

Mission is a strongly felt aim or goal in life, and is planned and executed with
zeal or dedication.

Plans are for fulfilment of selfish motives for the advancement of the objectives of individuals or a group.
Mission is a totally unselfish voluntary effort for the betterment of the life of individuals or groups unconnected to the missionary in any personal way.

A mission is an endeavour to achieve a specific goal. A plan is a set of instructions and intentions detailing how a mission or task should be accomplished.