The reason for the difference in translation has to do with the Hebrew word “Shachah”. The most literal meaning of Shachah is “to bow down”. But sometimes when someone bows down, they are not just showing reverence, but they are also engaging in worship. In other words, worship usually involves bowing of some sort, but not all bowing is worship. Sometimes bowing is just reverence.
The easiest way to illustrate this is when Moses did Shachah in two different occasions. In Exodus 18:7, Moses did Shachah toward his father-in-law so it is simply translated as “obeisance” which a synonym for reverence. However, in Exodus 34:8, when Moses did Shachah toward God – it was more than just reverence – it was an act of worship and it was translated as such.
Psalm 45 presents a dilemma for translators when it comes to translating Shachah. In this story, we see a King being married to the royal daughter of another king. But the story here is a prophecy of Christ being wedded to his church.
So, while the KJV would not normally translate someone bowing to a king as “worship” they decided to translate it here as worship and to capitalize the “L” in Lord to indicate it is a prophecy of Christ.
The ESV translated it as “bow down” instead of “worship” because while they agreed that this is a prophecy of Christ and his church that it is using a human story of a King marrying a royal daughter and therefore her bowing down would not be an act of worship, but rather an act of reverence.
I can see reasons for both translations. But in the end if we remember that marriage is a picture of Christ and the church than any reference to Christ and his bride has application to human marriage as well.
So, when applying this to human marriage and not the spiritual marriage of Christ and the church – Shachah must be understood in its most literal sense of bowing down in reverence.