According to Dictionary.com, "motivation" is "the state or condition of being motivated," or "something that motivates; inducement; incentive." One is motivated when they have a desire to do something such as pursue a goal or carry out some particular action. Motivation can come from any number of sources, including oneself or others. The incentives that motivate someone to do something can include personal gain, such as financial gain, or the desire to assist others and improve their quality of lives, such as with charitable work.
According to Dictionary.com, "achievement" is defined as "something accomplished, especially by superior ability, special effort, great courage," or "[the] act of achieving; attainment or accomplishment: the achievement of one's object." Achievement is thus used as a noun and refers to an accomplishment or successfully doing something challenging that one has set out to do.
"Achievement" therefore refers to the outcome itself while "motivation" refers to the reasons why someone would pursue that outcome in the first place.
Sometimes, however, desirable outcomes have nothing to do with motivation. For instance, mistakes or good luck lead to an outcome. But since there is no direct motivation in a mistake -- no one plans to make a mistake, by definition -- one cannot be motivated to make a mistake. Similarly, someone can enjoy a desirable outcome purely through luck, such as finding a suitcase full of money, without ever experiencing any real motivation.
In addition, people can be highly motivated toward achieving a goal, and work for years to reach that goal, but fail utterly to do so, falling short any sort of achievement whatsoever.
If someone wants to improve the living conditions of the local poor, she may volunteer her time to take on a project aimed at ameliorating poverty in her community. She may be motivated to do so by a desire to see individuals become happier. The happiness of others is thus the motivation for the individual. That individual may achieve her goal -- for instance, to build a shelter -- thanks in part to her motivation.