Martin Luther at the Diet of Worms in 1521: "Here I stand." From the 2003 film, Luther [source]
This topic is tough to concisely explain, as with all "large and lumpy" issues or inquiries. In a nutshell, I would say that there are big differences in the two approaches to Christianity and the Christian life.
Bear in mind, first of all, that whenever one is talking about "Protestantism," one must necessarily generalize, and there are always exceptions, since Protestants disagree with each other about virtually everything except the existence of God and that they are not Catholic.
The "Catholic mindset" above all, is the willingness to submit to the judgment of the Church and Tradition, as opposed to what is called "Private judgment" on the Protestant side, whereby the individual can dissent against Church and Council if he or she feels bound to do so by virtue of what they believe is the (contrary) teaching of Scripture.
Historically, one can date this approach from Martin Luther's "stand" at the Diet of Worms in 1521, when he appealed to Scripture alone and claimed that the Church and Councils "can and do err." That was a big turning point.
There are lots of other things. Many people have talked about how Catholicism approaches most theological matters with a "both/and" perspective, whereas Protestants tend to dichotomize and divide things and unnecessarily pit things against each other ("either/or"). Fr. Louis Bouyer, convert from Lutheranism, has called this a "dichotomous mindset." He wrote a wonderful book that discusses a lot of these worldview or presuppositional differences, called The Spirit and Forms of Protestantism. For more on this aspect, see my paper: Protestantism: Conceptual and Developmental Errors (section III).
Catholics are generally less individualistic, which goes along with the Protestant "mindset" of the individual and his Bible and His conscience and the Holy Spirit, etc. The Catholic always asks: "what does the Church teach about so-and-so?" "What has been the consensus of ages past and all the great saints and doctors and fathers of the Church?" Many Protestants (but not all, by many means) would think first and foremost, "what does the Bible teach about so-and-so?" Recourse to the Church and history of doctrine often never even crosses their minds. We appeal to the Bible, too, of course (ironically, and a bit humorously, on a Protestant blog, recently, I was basically chastised for citing too much Bible!). The difference is that we don't dismiss or discount the Church's say in the matter. And that is because we believe in an infallible Church. Protestants do not.
I've also noted many times how Protestants tend to have, in effect, a "priesthood of scholars" and that they are always endlessly searching for theological truths, as if some or many can never be obtained with certitude. They want (in many ways) to "reinvent the (theological) wheel" in every generation. In the epistemological sense this is almost a reduction of the Christian faith to mere philosophy, where the quest for truth is endless and never has a definitive stopping-point in a lot of areas of inquiry.
Catholics, on the other hand, believe that most if not all major theological doctrines have long since been resolved. They still develop, and there are remaining areas where we can disagree (plenty to talk about and mull over), but by and large we accept what has been passed down as true and feel no need to endlessly "quest" and "explore" for "truth" (or to relativize doctrines where Christians disagree, as if they are unimportant).
The following papers of mine delve into this "mindset" thing in one way or another, and from various different angles:
150 Reasons Why I am a Catholic
Why I Returned to the Catholic Church: Including a Searching Examination of Various Flaws and Errors in the Protestant Worldview and Approach to Christian Living (Al Kresta; edited and transcribed by Dave Armstrong)
Questions and Answers: Concerning Catholicism, Over Against Protestantism (Transcript of a Television Interview)
Response to Rev. Michael Pahls on "Theological Humility" and the Protestant "Non-Quest" Regarding Christian Certainty (vs. Anglican Rev. Michael Pahls)
Why I am a Catholic / Why I am a Lutheran / The Protestant Myth of Origins (vs. Lutheran Eric Phillips)
SOLA Scriptura vs. SOLO Scriptura, or, "Why History is Important For All Christians"
Protestant Ecclesiology and Epistemology is Always Ultimately Self-Defeating
400 Million "Popes": Protestant First Principles on Authority Are Inevitably Arbitrary, Unbiblical, and Viciously Self-Contradictory (+ Discussion) (vs. Baptist pastor Ken Temple)
Private Judgment: Its Meaning and How it is Viewed by Protestants and Catholics
My two moderator colleagues on the CHNI discussion board made some excellent observations in this regard. David W. Emery wrote:
Heresy is the undue emphasis of a certain portion or aspect of doctrine at the expense of another. Therefore, if the orthodox doctrine expresses its acceptance of “scripture and tradition” as the sources of revelation, heresy would demand that this be reworded as “scripture or tradition” and that one choose between them. The result is that “scripture alone” becomes one heresy, while “tradition alone” becomes another.Rick Luquette also elaborated along the same lines:
Another example is the orthodox concept of “faith and works functioning in harmony,” which in heresy is divided against itself as “faith or works, choose one.” Thus we have the “faith alone” camp and the “works alone” camp as the resulting heresies.
Let us consider a third example: “faith together with reason directing the whole man” is split apart by heresy into “faith alone” (fideism) and “reason alone” (rationalism), each with its militant adherents.
And a fourth: the tenet of “free will and predestination together” (actually a Catholic doctrine) is held to be contradictory, so it is demanded that “free will” be exclusive of “predestination” and vice versa. Thus we have, on the Protestant side, the Calvinists (predestinationists) in one camp and the Arminians (champions of free will) in another, splitting Protestantism in two.
The Catechism of the Catholic Church explains the corresponding Catholic position, which refuses to consider the terms contradictory but rather distinguishes more closely what is meant by “predestination”:
There is in fact such a thing as a Protestant mindset on the one hand and a Catholic mindset on the other.
CCC 600 To God, all moments of time are present in their immediacy. When therefore he establishes his eternal plan of “predestination”, he includes in it each person’s free response to his grace: “In this city, in fact, both Herod and Pontius Pilate, with the Gentiles and the peoples of Israel, gathered together against your holy servant Jesus, whom you anointed, to do whatever your hand and your plan had predestined to take place.” (Acts 4:27–28; cf. Psalms 2:1–2) For the sake of accomplishing his plan of salvation, God permitted the acts that flowed from their blindness. (cf. Matthew 26:54; John 18:36; 19:11; Acts 3:17–18)
From Genesis to Revelation, we read of the "woman". In Genesis, she crushes the serpent's head; in Revelation she wears a crown of 12 stars. Protestants will question whether the woman represents the Church or Israel. Some might even ask if the woman is Mary.
A Catholic will say that the woman is all three. Our viewpoint tends to be broader. Instead of there being one meaning, we see many, and recognize them all as correct.
Protestants tell us that we should not use a crucifix because Jesus is no longer on the cross. Catholics see the sacrifice of the cross as a timeless sacrifice that is eternally being offered, and is made present again in the mass. Protestants tell us we are crucifying Jesus all over again; we say we are making the eternal sacrifice present once again in the Eucharist. We are re-presenting the same eternal sacrifice offered once for all.
Protestants say how can it mean this and that at the same time? We say it can and it does. God encompasses all truth. Thus Jesus can tell us Christians that baptism is necessary for salvation, and still offer salvation to those who, through no fault of their own, have never known of their need to be baptized. Protestants will say that Catholics believe we are saved by our works; Catholics say that our works demonstrate our response to our salvation.
We abbreviate it by saying that the Protestant mindset is "either/or" while the Catholic mindset is "both/and".