What they share
Both are four-syllable classical names with deep historical roots. Both offer multiple nicknames (Alex, Xander, Al for Alexander; Ben, Benji, Benny for Benjamin). Both are universally readable across languages.
Pick Alexander if…
Pick Alexander if you want the Greek, warrior-king heritage. Alexander means 'defender of the people' and carries the weight of Alexander the Great. Alex as a short form is confident and unisex-adjacent.
Pick Benjamin if…
Pick Benjamin if you want the Hebrew, familial warmth. Benjamin means 'son of the right hand' and carries biblical tenderness. Ben as a short form is simple, classic, and ages from infant to elder without strain.
Alexander is a shade more ceremonial; Benjamin is a shade more affectionate. Both work at every age. The nickname you would actually use day to day often decides: Alex versus Ben is a good question to sit with.