The world of online marketing changes quickly. Over the years, terminology used to discuss both ways in which people market themselves or their organizations to search engines. In an effort not to re-invent the wheel, I think Hubspot’s article is a good starting place.
In practice, SEO (search engine optimization) has been used to describe practices to improve a website’s search rankings (originally primarily using keywords in titles and metadata but more and more having quality content, fast-loading pages, well-formatted URLS, quality links to your site, etc).
SEM (search engine marketing), which has at times been used by businesses and agencies to describe paid search (which includes but is not limited to PPC) exclusively. However, as more and more organizations use paid search, the term has grown to encompass both paid search and SEO strategies. Language evolves over time and there’s always going to be some splitting of hairs, but SEM is generally agreed-upon to be comprised of both paid and unpaid strategies and includes SEO, which only uses unpaid strategies.
The terminology might change over time (for instance, is using Structured Data in your website code SEM or SEO?) but whether you’re working on organic SEO or increasing visibility/traffic via SEM, an important part of your goal is influencing what appears on the SERP (search engine results page).
Both SEO and paid search matter because while the discerning internet user knows which results have been paid for, if one were in a hurry to say, find Chinese food in New York, one might be tempted to click on the groupon.com ad. This is backed up by research that found that “[c]icks on paid search listings beat out organic clicks by nearly a 2:1 margin for keywords with high commercial intent in the US.“
Anyways, I hope that helps.