If a non- Jewish cashier forgot to scan an item, do I have to go back and make sure they charge me for it?
The Avnei Yoshfei has a teshuvah on this subject.
Regarding a non-Jew from whom one bought 4 items and who charged only for 3, he says that this is טעות עכו״ם and is allowed. The Jew does not have to tell the non-Jew that he is relying on the non-Jew's account.
Halacha 4 A mistake in accounts made by a gentile is considered like a lost article, and it is permissible to benefit from it. This applies if he makes the mistake himself. It is, however, forbidden to cause him to make a mistake.
Halacha 5 What is implied? A gentile made an account and made an error in a Jew's favor. The Jew must tell him: "Look, I am relying on your account. I know only what you tell me." In an instance like this, if the gentile does not correct himself, it is permitted to take advantage of his error. If, however, one does not tell him this, it is forbidden. It is possible that the gentile's intent is to check the Jew's honesty. By keeping the money, one may cause God's name to be desecrated.
Further business halacha.com says that
if the Jew returns the error to sanctify Hashem's name so that non-Jews will praise us and say we are faithful in business, this is praiseworthy. אם החזיר טעות עכו"ם לקדש את השם כדי שיפארו את ישראל וידעו שהם בעלי אמונה, ה"ז משובח
Unless the amount is de minimus (of minor importance), then it is an opportunity to make a Kiddush Hashem. Passing over that opportunity is throwing away a Mitzvah, and if caught could lead to a very bad Aveirah. I'm very skeptical that you could find a heter for that. This question discusses it.
That is really the distinction between a Jew and a non-Jew here - for a Jew, even if the amount isn't enough to make a big deal about and the person wouldn't care very much, you have to go to the effort of returning it. For a non-Jew, it more depends on their perception of the situation. A lot of discussion in these Halachos focus on the details, which are important, but tend to elide over the common sense points that apply generally because they are taken for granted and are not tied to the particular facts of this situation.