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Let's say you're running a marathon and someone catches up, runs faster than you, and wins the run at the last minute.
Did the person who catch up to you and won the marathon passed you while you were running or did they cut you off and passed you?
Does pass alone explain the idea that the person ran faster than you and won the race or do I have to have cut me off after that to sound more natural and explain the action?
No. "Cut off" implies an action, often deliberate but not necessarily, that causes an interruption to you or interferes with you in some way.
In your marathon runner example, cutting off would suggest one runner veering to be directly in front of another so as to force the other to have to adjust pace to avoid collision or tie up with the first. I am more familiar with the term in a motor vehicle traffic context. A driver who cuts off another is one who changes lanes suddenly, forcing the other to brake heavily (and to sound the horn and make various gestures.). Simply passing another car, however, means the first driver waited to change lanes until well ahead of the slower car.
Edits (7 May 2019): Since creating this answer a few weeks ago, I've had a couple thoughts that may help further.
First, in a remarkable case of "Life imitates the Stack Exchange," the results of the 2019 running of the Kentucky Derby provide us with some excellent illustrations of the concepts here. (I will describe here things as called by the stewards, debate on such is way beyond scope here.) During the race, horse Maximum Security cut-off horse War of Will. The rear legs of Maximum Security and the front legs of War of Will struck together, forcing both horses to alter stride, signicantly impacting War of Will's position in the race. Meanwhile, horse Country House benefited from this, using its position on the outside to pass several other horses and eventually become the official winner.
Second, it occurs to me there is another idiom that the OP may have confused with cut off, that being caught up. In verb definition 1b at Merriam Webster, the meaning of "to reach a state of parity" is the one intended here. It suggests coming from behind to be much closer or nearly equal. Although normally phrased as "caught up to me" in some dialects this can be "caught me up," which then follows the verb-me-preposition pattern: cut me off versus caught me up. When spoken rapidly or with certain accents, these could sound similar and be confused with each other.
To summarize, I think it helps to think of the differences in these terms in before/ after contexts:
