The most prominent similarity is that they both serve the purpose of convincing the audience about a certain point. Speeches rely more on tone, while essays rely more on content due to the large amount they can fit in.
Similarities of persuasive speeches and persuasive essays:
Both communicate information to their <em>live</em> or <em>reading</em> audience.
Differences of persuasive speeches and persuasive essays:
Persuasive speech is delivered to a live audience it needs to be more descriptive and entertaining, presenting a tone that produces an emotionaleffect on the audience. At the end od each section, it's important to <em>restate</em> the point.
Perusasive essay: in general it is used to comunicate to a general audience. At the conclusion, the information is <em>synthesized</em>.