I’m a full blown TULIP believer myself, but not as educated or eloquent as the other Reformed posters on this thread. I only have a few things to add.
First, the key difference in my mind between Reformed thinking (TULIP theology) and cooperative faith, either Catholic or other denominations like Methodist, Baptist, etc, boils down to God’s sovereignty.
Either God is completely sovereign, and His will is all that comes to pass, or God somehow allows His own will to be thwarted by His creation. In cooperative theology, Christ’s death was sufficient for all to come to God, yet only those who choose to are saved unto life. This means that God’s will is thwarted by the millions of souls who chose to reject Him. This renders man at least as powerful as God – only we are allowed to choose. God can only make the offer and hope we’ll come to Him. The same argument follows for Perseverance of the Saints as with Irresistible Grace. Either we can cooperate with God or we can block His will. God is reduced to a sideline cheerleader almost pleading with us to take His offer. This is not the God I see explained in the scriptures.
Secondly, about the Fall; the most glorious thing about being Reformed is the knowledge that God meant for Adam to fall – it was His express will. It had to be, or the Elect, known before time, would not have been necessary. If there were some chance that Adam would not sin, then all might have been able to live in Eden. Out of that, God changed us from created people – made from dust and bone – into sons of God, joint heirs with Christ. We are brothers to the Only Begotten Son of the Father! All because of the fall and the Redemption that God planned from the beginning.
Adam, before the fall, was only a created being. Adam, after the establishment of Christ’s church outside the garden, was a son of God; he was made equal to Christ through the imputation of Christ’s righteousness.