The Christian Duck Test? If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.

There is arguably little difference between the lifestyle of the Stoic and the lifestyle of the Christian. Both eschew effeminacy in pursuit of virtue. Both find it best for man to worry about things he can control and to not worry about the things he cannot (Epictetus wrote of this. Christians call this &quot;trusting the plan&quot;; there is also the Christian serenity prayer which expresses this). Both believe in the cardinal virtues of Prudence, Justice, Fortitude, and Temperance (Christians certainly ought to believe these, and I'd bet most Stoics do believe them as well). I would argue that their way of life is in essence, the same.
The question is this: if a Christian and a Stoic both live the same exact life, and the only difference in their essence is that the Christian believes in the gospel, then why does the Christian go to heaven while the unbelieving Stoic does not? More generally, the question is about the &quot;mechanics&quot; of Christian salvation: when one comes to believe in Christ (esp. when converting from stoicism) what is it that actually changes in the person's essence/soul/existence that makes him fit for the Kingdom of God? In other words, what is it about the intellectual position of being Christian that impacts the state of someone's soul?
If we assume that the Christian worldview is accurate, then a true Christian has something that a Stoic doesn't: the Holy Spirit.
Acts 1:8 (ESV):

8 But you will receive power when the Holy Spirit has come upon you, and you will be my witnesses in Jerusalem and in all Judea and Samaria, and to the end of the earth.”

The Holy Spirit empowers the Christian to be able to overcome the desires of the flesh and produce the fruit of the Spirit:

16 But I say, walk by the Spirit, and you will not gratify the desires of the flesh. 17 For the desires of the flesh are against the Spirit, and the desires of the Spirit are against the flesh, for these are opposed to each other, to keep you from doing the things you want to do. 18 But if you are led by the Spirit, you are not under the law. 19 Now the works of the flesh are evident: sexual immorality, impurity, sensuality, 20 idolatry, sorcery, enmity, strife, jealousy, fits of anger, rivalries, dissensions, divisions, 21 envy, drunkenness, orgies, and things like these. I warn you, as I warned you before, that those who do such things will not inherit the kingdom of God. 22 But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, 23 gentleness, self-control; against such things there is no law. 24 And those who belong to Christ Jesus have crucified the flesh with its passions and desires. [Galatians 5:16-24, ESV]

The Holy Spirit also distributes supernatural gifts among Christians as He sees fit:

4 Now there are varieties of gifts, but the same Spirit; 5 and there are varieties of service, but the same Lord; 6 and there are varieties of activities, but it is the same God who empowers them all in everyone. 7 To each is given the manifestation of the Spirit for the common good. 8 For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills. [1 Corinthians 12:4-11, ESV]

In light of this, when you say &quot;if a Christian and a Stoic both live the same exact life, and the only difference in their essence is that the Christian believes in the gospel [...]&quot;, I say: the condition of your question is ill-defined from the Christian viewpoint, since a Stoic, who lacks the gift of the Holy Spirit, by definition, cannot be living the same exact life as a Spirit-filled Christian. The latter enjoys the supernatural empowerment and assistance of the Spirit of God that the former lacks.
I will close with a pertinent quote from Romans, chapter 8 (ESV):

9 You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him. 10 But if Christ is in you, although the body is dead because of sin, the Spirit is life because of righteousness. 11 If the Spirit of him who raised Jesus from the dead dwells in you, he who raised Christ Jesus from the dead will also give life to your mortal bodies through his Spirit who dwells in you.
12 So then, brothers, we are debtors, not to the flesh, to live according to the flesh. 13 For if you live according to the flesh you will die, but if by the Spirit you put to death the deeds of the body, you will live. 14 For all who are led by the Spirit of God are sons of God. 15 For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of adoption as sons, by whom we cry, “Abba! Father!” 16 The Spirit himself bears witness with our spirit that we are children of God, 17 and if children, then heirs—heirs of God and fellow heirs with Christ, provided we suffer with him in order that we may also be glorified with him.


Answering concerns raised in the comments:
(1) Is the spirit something you can feel within you, and recognize in others? Also, regarding faith, do you &quot;know&quot; when you have faith in Christ? Is it a change in category or a change in degree?
I think these questions can be answered both scripturally and by appealing to the first-hand experience of many believers.
Scripturally, the Bible has lots of examples where believers were filled with the Spirit and had extraordinary experiences. The book of Acts is a good example. That said, a more common --and perhaps less extraordinary-- experience is the &quot;inner witness of the Holy Spirit&quot;, which is described very well by Romans 8:15-17 (already quoted above):

15 For you did not receive the spirit of slavery to fall back into fear, but you have received the Spirit of adoption as sons, by whom we cry, “Abba! Father!” 16 The Spirit himself bears witness with our spirit that we are children of God, 17 and if children, then heirs—heirs of God and fellow heirs with Christ, provided we suffer with him in order that we may also be glorified with him.

Experientially, throughout history many believers have reported personal encounters and extraordinary experiences with the Spirit of God. I have asked questions about this in the past, feel free to take a look: 1, 2, 3.
(2) This answer is self contradictory, since at least some of the fruits of the Holy Spirit listed therein coincide with Stoic virtues, yet the answer goes on to subsequently deny that the Stoics are, at least partially, indwelled by God's Spirit.
Without the Holy Spirit, a Stoic would not be able to achieve 100% of what they otherwise would be able to with the power of the Holy Spirit. But that doesn't mean that they will achieve 0%. It's still theoretically possible for a pagan to develop some moral virtues that would place them somewhere in between the two extremes. Claiming that it's either 0% or 100% would be a false dichotomy, and I never intended to claim such a thing in my answer.
Regarding the salvation status of virtuous pagans, this question might be of interest: What is an overview of Christian viewpoints on the eternal destiny of individuals who die never hearing the Gospel?