Argumentative and persuasive essays are similar in that they both are about an issue on which even experts may differ. In other words, for example, you cannot write an argumentative essay on the topic of whether the world is roughly spherical or whether 2+2 equals four as those are not topics on which there are differing opinions, but you can write persuasive or argumentative essays about social policy or ethics.
Although the two types of essay are similar in content, generally when instructors assign them, there is a slight difference in focus. An "argumentative" essay focuses on using different types of logical proof or evidence to support a point; the emphasis in grading such an essay is on logical coherence and how well the essay succeeds in supporting its thesis. The term "persuasive" essay indicates an emphasis on audience, and is evaluated in terms of how effective the essay might be in convincing its audience of its central points.
It is possible to write both persuasive and argumentative essays advocating a proposed solution to a perceived problem, and thus some essays in these genres are problem-solution essays.