Both are the real thing
There's no trap here, which is the first thing to say. The Pro V1 and the Chrome Soft are both genuine premium golf balls with soft urethane covers and multi-layer construction - the tier a serious golfer plays and rations. Giving either to a golfer is a good gift. This is not a case where one option is a mistake.
That means the question isn't "which is better" in the abstract. It's which one suits the specific person, and for a gift buyer the answer is almost always decided by loyalty rather than by specs.
Why brand loyalty decides it
Golfers who play premium balls tend to have a brand they stick with, and the difference between the two at this level is smaller than the difference in what they're used to. A Pro V1 player notices immediately if you hand them a Chrome Soft, and vice versa - not because one is worse, but because feel around the greens is personal and they've calibrated to theirs.
So the highest-confidence move is to find out what they play and match it exactly. Look in the ball pocket of their bag or the empty box in the bottom; whatever brand is there is your answer, and matching it needs no golf knowledge at all.
How they differ, honestly
If you want to choose on character rather than loyalty, the general reputation is this: the Pro V1 is the benchmark, the most-played ball in professional golf, with a firm-but-lively feel. The Chrome Soft leans, as the name says, toward a softer feel while still performing at the tour level. Both spin and stop well for skilled players; neither is a distance ball built purely for yardage.
These are reputational generalizations from how the manufacturers position the balls, not lab measurements - we don't test equipment, and how we pick says so plainly. For a gift, the character difference matters far less than the brand match.
Which to buy as a gift
If you know their brand, buy that. If you genuinely don't, default to the Pro V1 - it's the most-recognized premium ball in golf and the safest "I bought you the good ones" gift there is. Choose the Chrome Soft instead if you know they play Callaway clubs or have mentioned liking a soft feel. Either way, a dozen premium balls is a gift a golfer is glad to get, because it's exactly the thing they hesitate to buy at full price for themselves - more on that in are expensive golf balls worth it.