Speaking as a Protestant, I suggest that one of the main "sticking points"--if not the sticking point--between the Protestant and Catholic perspectives on the personhood of Mary can be traced to our differing perspectives on Holy Scripture.
Generally speaking, Protestants (and particularly Evangelical Protestants, of whom I am one) consider the Holy Bible to be the only fully authoritative word of God. Catholics, on the other hand, while having great reverence for the Bible, do not consider the Bible to contain the only fully authoritative word of God. What is fully authoritative to the Roman Catholic Church is church tradition as interpreted by the leadership within the Roman Catholic hierarchy, and particularly by the Holy See, and the authority, jurisdiction, and governmental functions associated with the papacy.
Speaking more specifically, Waeshael's answer, above, contains a good illustration of how a scripture is interpreted through the lens of church tradition. I quote his answer as follows:
On the Cross (John 19:26) the Master told this to a disciple: “When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son! Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home” (John 19:26, 27 KJV). The Church Fathers understand this command to be directed towards all who believe that Jesus loves them. To be considered brethren to Jesus, you shall consider Mary to be your Mother, and you must take her into your own home.
Your average Evangelical Christian would interpret the same verses quite differently, I suggest. First, our (i.e., my) interpretation would involve very practical--nay, pragmatic--concerns such as John's support of Jesus' earthly mother, not only in the wake of Jesus' immediate demise on the cross, but also after Jesus' ascension into heaven.
Second, we would suggest that Jesus' attitude toward his earthly mother was at times a bit less than reverential. When Jesus was preaching to the crowds in his hometown of Nazareth, someone told him,
"Behold, Your mother and Your brothers are standing outside seeking to speak to You" (Matthew 12:47; cf. Mark 3:32 and Luke 8:20).
What was Jesus' response?
"But Jesus answered the one who was telling Him and said, 'Who is My mother and who are My brothers?' And stretching out His hand toward His disciples, He said, 'Behold My mother and My brothers! For whoever does the will of My Father who is in heaven, he is My brother and sister and mother'" (Matthew 12:48-50 NASB).
Perhaps a Catholic contributor to this website would care to interpret that passage from a Catholic perspective, as I'm sure his or her interpretation would differ significantly from that of most Protestants.
In conclusion, Catholics, I suggest, are simply not bound as tightly to the Judeo-Christian Scriptures as are many--if not most--Protestants. I would not be surprised if many Catholics interpreted our reverence for Scripture as bordering on Bible worship--or bibliophily at best. I have to admit, many famous Protestant preachers seem to have a habit of iterating the same "The Bible says . . ." ad infinitum, ad nauseum.
Moreover, Protestants are not unlike many Catholics, I suspect, in realizing just how much our common religion has been shaped by and depends on centuries of tradition and generations of socialization regarding what we believe and why we believe it! What is most important, in my opinion, is that at the heart of our common religion is a living Lord and Savior, Jesus Christ. That alone should provide sufficient common ground for cordial relations between us.