If you ask a Protestant what the Church is, they'll probably give a variety of answers, although one of the most common is "all Christians"--this answer, unfortunately, has no Scriptural basis, especially since the application of the term "Christian" has widened considerably over the last two thousand years. Although if we narrowed the definition considerably, then it would be another story.
However, I don't blame Protestants entirely for this, since I can see from the Catholic Catechism that Rome has a likewise humanist understanding of the Church, perhaps that is where Protestantism got it. In defining the Church, the Roman Catechism states,
752 In Christian usage, the word "church" designates the liturgical assembly, but also the local community or the whole universal community of believers. These three meanings are inseparable. "The Church" is the People that God gathers in the whole world. She exists in local communities and is made real as a liturgical, above all a Eucharistic, assembly. She draws her life from the word and the Body of Christ and so herself becomes Christ's Body.
Now it is clear that Rome does understand the Church as the Body of Christ (and so do Protestants), but instead of proceeding from that definition, she ends with it. Which is a problem. Scripture starts from that definition, the Church "is his body, the fulness of him that filleth all in all." (Ephesians 1:23). Rome starts with the Church as something external to God, which
becomes the Body of Christ. In Orthodoxy, the fundamental definition of the Church is God's Body and Fullness. The transformation is rather of the people being united to the Church. This perversion in definition engenders a radically different Roman theology in general, for instance, also from the Catechism,
882 The Pope
, Bishop of Rome and Peter's successor, "is the perpetual and visible source and foundation of the unity both of the bishops and of the whole company of the faithful." "For the Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered."
Now in Orthodoxy, clergy preside in the Church, but the idea of clergy having power
over the Church would be sacrilegious, in fact arguably blasphemous.
Further down the line, Catholic ecclesiology seems to have created the idea that the
Church can sin. Cardinal Marx, for instance, recently said the Church should apologize for not having been supportive enough of gay rights:
Cardinal Marx: Homosexuals deserve an apology from the Church
Now I'm not going to address how inane that is, but I will point out that in Orthodoxy, sin is precisely stepping away from the Church, a sundering from Her. Repentance and Communion as are a rejoining and a repairing of the damage.
The most potent of this dreadful theology can also be found in the Catechism. Concerning the Church, it is said:
Here below she knows that she is in exile far from the Lord [769]. This is simply heretical in the highest degree, the Church is the Lord's Body and Fullness, and in joining with it, we become One with him--how is that exile?
So while the Pope and the Filioque might have engendered the Great Schism, it is clear that they are hardly the only things that separate us today. Catholic theology has diverged enormously from Orthodox theology, and saying there are cosmetic similarities like clergy, doesn't mean there is necessarily a lot of common ground.