The main difference is that Search Engine Optimization (SEO) is focused on optimizing a website in order to get traffic from organic search results. On the other hand, the goal of Search Engine Marketing (SEM) is to get traffic and visibility from both organic and paid search.
Put another way:
Google’s search results are divided into two main categories: the paid search results and the organic search results.
The goal of SEO is to rank your website in the organic search results.
You can also get your website in the paid area of the search results via pay per click (PPC).
SEO is where you focus 100% on ranking in the organic results. SEM is when you tap into both SEO and PPC in order to get traffic from search engines.
So yeah, SEM is a broad term that includes SEO and PPC. Which means that SEO falls “under” the umbrella category of SEM.