The 14th amendment says: "No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; ... nor deny to any person within its jurisdiction the equal protection of the laws." But up until the 1960s at least, most white Americans did not want equal protection - they wanted substantially inferior public accommodations and treatment for non-whites, bans on non-whites from marrying whites, no black police officers, etc, etc, etc, etc. So how did the amendment pass without substantially different wording?
This is tied to the Civil War where half the States strongly supported slavery and half strongly opposed slavery. The fourteenth amendment was passed soon after the Civil War. At the start of the war, Southern States quit the federal Congress and tried to form their own country. In losing the war, they failed to have their own country based on the preservation of slavery. They were not immediately readmitted into Congress. So, they were run by the North until they completed some reforms. The winning Northern States passed three amendments to give rights to the freed slaves. They didn't allow the southern States to rejoin Congress until they voted to approve these amendments.
Watch the movie Lincoln which came out in 2012 if you want to see a movie on the general topic. That was the thirteenth amendment but the 14th and 15th were passed in similar manner.