The context of the passage of Paul's letter to Timothy continues...
Prescribe and teach these things. 1 Tim 4:11 (NASB)
Earlier in the same epistle Paul wrote that Christ Jesus gave Himself as a ransom "for all" (1 Tim 2:6). Therefore it is the desire of "God our Savior" that "all men" be saved (1 Tim 2:4 and 2 Pet 3:9). The Apostle John says that Christ Jesus died for the sins of the whole world, and not just for the sins of believers (1 Jn 2:2), and so the writer of the Book of Hebrews indicates that Christ Jesus tasted death "for everyone" (Heb 2:9).
These verses are explicit, independent propositional statements. In order to avoid doctrinal interference one must turn to Romans 5 to understand why believers are "especially" saved.
Romans 5 juxtaposes Adam with Christ Jesus. That is, the first Adam disobeyed God in the Garden of Eden, and the second Adam obeyed God in the Garden of Gethsemane. The difference was that one man (first Adam) was a living soul (1 Cor 15:45) and the other man (second Adam) was a life-giving spirit, who existed as eternal life (1 Cor 15:45).
So the first Adam catapulted the entire human race into spiritual separation from God through his disobedience (please click here). All human beings are born into the state of spiritual death, and this is Problem #1. The result was that all descendants of the first Adam became transgressors against God (please click here). That is, all human beings commit transgressions as individual acts of disobedience, and this is Problem #2.
The second Adam comes along and dies for the sins of the entire world. He dies for the sins of the entire world because he is not under the curse of spiritual death. He was born with eternal life, because he existed as Eternal Life. In addition, he never sinned (2 Cor 5:21), and so he never committed any transgressions against God or man. As the second Adam he absorbs the transgressions of the world in his own body (please click here). So as the second Adam, he tastes death "for all men," since he dies for the sins of the entire world (1 Jn 2:2). The sacrifice on the cross resolves Problem #2 for "all men."
Now where believers are "especially" saved comes to receiving the free gift of eternal life (please click here). That is, the believer is identified as one who is "born again" with eternal life (Jn 3:5-7). The message of the gospel is that Christ died for all men, and that the sinner can receive righteousness through faith, which results in the free gift of eternal life. The good news is that one can be free from not only the condemnation of sin, but the power of sin. Thus believers have Problem #1 resolved for them. This does not mean that believers become sinless people, but that they do not "practice" unrighteousness (1 Jn 2:29; 1 Jn 3:10; Rev 22:11; inter alia). Remember: we still remain the biological descendants of the first Adam, and therefore our bodies still retain the Adamic nature, but the believer begins to practice righteousness as a slave of righteousness...
The preceding four paragraphs have thus condensed the main thrust of Romans 5, which is to show how believers are "especially" saved.
The unbeliever at the last judgment therefore is not condemned to eternal damnation because the "books of sins" were opened, but because the "books of (dead) works" were opened. In other words, sins are not the basis of the indictment at the last judgment, but the dead works of personal righteousness, which amount to spiritual death -- that is, it is Problem #1 that is what gets them thrown into the Lake of Fire. It is spiritual death and its entourage from Hades (all unbelievers) which are thrown into the Lake of Fire at the last judgment (Rev 20:14), and therefore is not for the condemnation of sins.
Sins of the ENTIRE world were condemned and nailed to the cross (Col 2:14), and thus Problem #2 was resolved for "all men." (Problem #1 and Problem #2 are mentioned together in this immediate context in Colossians 2:13.) Thus there is no unbeliever who can claim that his sins are unforgivable, or that his sins have disqualified him from God's eternal election to the saved. Therefore all sinners can be saved. It is now a question of being called by faith.
So the issue is not, "are you a sinner?" because all men are sinners and have fallen short of the glory of God (Rom 3:23). The issue instead is, "Do you have the righteousness --and therefore the free gift of eternal life-- through faith on the substitutionary atonement of the Lamb of God, the ATONEMENT of the sins of the world?"
Thus the believer is "especially" saved because he has righteousness and eternal life. What he shares with the unbeliever is that all sins of all people of all times were nailed and judged on the cross. But the believer is "especially" saved because he believes by faith in the substitionary atonement, which is what results in righteousness and the free gift of eternal life (and therefore the elimination of spiritual death). In other words, both the believer and the unbeliever have Problem #2 resolved for both of them through the cross of Christ Jesus (and thus all are qualified for salvation), but only the believer has Problem #1 resolved, and so he is "especially" saved.
In summary, because of doctrinal interference, these nuances of truths in Romans 5 have been obfuscated over the centuries, and so there has been much confusion as to why believers are "especially" saved.