Marketing and public relations differ primarily in their focus.
The objective of marketing is to drive profit. You are communicating with people in an effort get them to buy your product.
In public relations you are also communicating with people, but instead your focus is managing the client's image.
(Hence why both marketing & PR fall under communications, they are both functions of communications but the goal is what seperates them)
You may be targeting the general public, sometimes the shareholders of a public company (Investor Relations) and sometimes the employees of your client (internal public relations)
For both marketing and public relations, a commonly used medium of getting the message across is advertising.
A single medium can be used for multiple purposes, in this case both marketing and PR. In marketing, the advertisement's goal is to generate sales / profit, while in PR, this same medium is used to manage public image.
You can have a banner advertising a new sales event at a department store one month, and then a banner at the same department store promoting their charity 5K race the next month. Both of these campaigns are using advertising as a medium to get their message across, but it is the goal of each method that differentiates the marketing vs the PR function.
As mentioned earlier, both marketing & PR fall under communications, which is the practice of distrbiuting and receiving information.
Some things to remember:
According to Cutlip & Center's Effective Public Relations, PR can be a part of a marketing strategy, while marketing cannot be a part of a PR strategy.
In other words, the way your brand is perceived (a PR responsibility) can be a crucial part of driving profits (a marketing responsibility) but not the other way around.
This is because driving profits is exclusively part of marketing, PR can help drive profits, by making sure a company has a positive public image, and ultimately help the marketing function, but driving profits is usually not a part of garnering a positive public image.
Many companies use the methods of PR to support marketing and sales objectives, which is referred to as marketing communications or marketing PR.
Sources:
Broom, Glen M., et al. Effective Public Relations. Prentice Hall, Pearson, 2009.
Kelleher, Tom. Public Relations. Oxford University Press, 2018.
