I'm not sure how accurate this post will be. I'm neither a religious nor socio-cultural expert. I am just stating and analyzing observations that I have made throughout the years.
Before meals, I always pray. Depending on who I am with, I pray differently. Whenever I'm with my nephew, we sing the Prayer Before Meals song:
"God is Great and God is good,Whenever I'm alone or with my parents, we pray the prayer before meals that I learned when I was young and they learned when they were young:
Thank you, God, for our food."
"Bless us, O LordWhen I am with my non-Catholic Christian friends, we don't use these prescribed prayers that Catholics commonly use. With my friends, our prayers are a bit more organic, more customized to our specific needs, so to speak. Although we follow the "ATS format" (Acclamation-Thanksgiving-Supplication, no Confession/Contrition, because the prayer leader cannot be contrite for the sins of his/her companions), the prayers we pray as friends are rarely the same.
And these, Your gifts,
Which we are about to receive
From Your bounty
Through Christ, our Lord
Protestants  do not have "memorized prayers". Now, I don't believe that there is a right way and a wrong way of praying. I believe that a prayer, memorized or not, so long as it is spoken from the heart, is still the most effective way of communicating with God. But, when faced with the question why Catholics must pray memorized prayers, I found myself giving my own theory on the matter.
Catholics focus on the community. We are constantly reminded that we are part of the "Body of the Church with Christ as our Head". With our memorized prayers, Catholics can echo the prayers of other Catholics from other parts of the world. We "believe in the Holy Catholic Church" as much as we believe in "God, the Father Almighty", "Jesus Christ, His only Son, our Lord", and the Holy Spirit. On a less spiritual level, I attribute this focus to the age of our religion. Catholicism, being the oldest Christian religion, was born at a time when men focused on the community. On the other hand, religions formed during the Protestant Reformation  focus on the personal relationship of the individual with God. I am often asked if I have accepted Jesus Christ as "my personal Lord and savior". As we do not memorize litanies to repeat them in different conversations over and over again, so Protestant brothers and sisters do not memorize prayers.
I was in high school when we read Homer's Iliad and my English teacher said that "epics can no longer be written in this age of novels" (I think she was quoting someone). People are now more focused on their own philosophies and ideals. When writers write, it is of their personal thoughts. Once upon a time, man could not separate himself from his community. When a writer wrote, he wrote about the philosophies and ideals of his community or race. The differences in the literary styles of the eras show the differences in the mindset of the people. The differences in the mindset of the people who established the different religious institutions created the differences in prayer practices. This is, of course, only my humble opinion.
Please allow me to make it clear that we Catholics do not ignore our personal relationship with the Father, Son, and Holy Ghost. Similarly, though it is not their main focus, Baptists also believe in the church community (else, why would there be such a thing as Baptist religion?). There are just different focuses, and therefore, different practices and approaches towards the one goal of salvation.
 I really hate that word. It's so medieval. Unfortunately, it is the only term I can think of at the moment.
 wiki said "The Protestant Reformation was a movement which emerged in the 16th century as a series of attempts to reform the Roman Catholic Church in Western Europe. The reformation ended in division and the establishment of new institutions, most importantly Lutheranism, Reformed churches, and Anabaptists. It also led to the Counter-Reformation within the Roman Catholic Church."