There are a lot or reasons that one could give for these differences. The truth is that in the 13th Century, Innocent III, Bishop of Rome, decided black would be the color for penitential seasons. Unfortunately in those days, it was very hard to get dyes to work just right. This meant sometimes the color was a little off or that it would fade very quickly. This is likely what led to the use of blue in England with the Sarum Rite of worship and purple on the continent of Europe. It's not that there is any hidden meaning with one or the other, it's just that blue and purple were the closest colors to black people could get.
This is helpful for us to realize today. Both Advent and Lent are Penitential Seasons in the church. They are both preparations for the two major feasts of the church, Christmas and Easter. Using purple for both can emphasize that.
But they are different too. Lent prepares us in moving forward to the feast of Easter. It reminds us of the past, although it can be a sign of moving forward in our life in faith if we are preparing for Baptism, Confirmation, Reaffirmation, or Reception at this time.
Advent, on the other hand, is not just about preparation for the feast of Christmas, and it's not just about remembering the events of the past. It's about preparing for the events of the future as well. In The Nicene Creed, we remind ourselves of the time when Jesus will "come again in glory to judge the living and the dead". We are preparing ourselves in Advent, as Christians have since before Christmas became a major feast of the church, for Jesus' return, of which we don't know the how, when, or where of yet. So while Advent prepares us for the feast of Christmas, it also prepares us for the future.
If we want to make this distinction between Advent and Lent, then by all means use blue. Jesus will be fine with whatever choice we make. But we should also look at the history of these two colors to realize the similarities between Advent and Lent as well.
Sources used:
Alexander, Neil. Creating Liturgical Time. New York: Church Publishing, 2014, pg. 51.