To understand the relationship of sin (παράπτωμα) to sin (ἁμαρτία) I’ll spend some time considering Paul’s usage of the words in other writings.  Both words were used in Romans 5:12-15 (NET).

So then, just as sin (ἁμαρτία) entered the world through one man and death through sin (ἁμαρτίας, a form of ἁμαρτία), and so death spread to all people because all sinned (ἥμαρτον, a form of ἁμαρτάνω) – for before the law was given, sin (ἁμαρτία) was in the world, but there is no accounting (ἐλλογεῖται, a form of ἐλλογέω) for sin (ἁμαρτία) when there is no law.  Yet death reigned from Adam until Moses¹ even over those who did not sin (ἁμαρτήσαντας, a form of ἁμαρτάνω) in the same way that Adam (who is a type of the coming one) transgressed (παραβάσεως, a form παράβασις).  But the gracious gift (χάρισμα) is not like the transgression (παράπτωμα).  For if the many died through the transgression (παραπτώματι, a form of παράπτωμα) of the one man, how much more did the grace (χάρις) of God and the gift (δωρεά) by the grace (χάριτι, a form of χάρις) of the one man Jesus Christ multiply to the many!

Paul made a very specific differentiation between ἁμαρτία and παράπτωμα in this passage, but it is not one of greater or lesser degree.  Sin (ἁμαρτία) entered the world through one man’s παράβασις, that was Adam’s transgression of a specific command not to eat of a particular fruit.  Death entered the world through sin (ἁμαρτία).  Death spread to all people because all sinned (ἁμαρτάνω) despite the fact that not all sinned by breaking a specific commandment like Adam had.  The clause, For if the many died through the transgression (παράπτωμα) of the one man, links παράπτωμα to Adam’s παράβασις.  So sin (παράπτωμα) is the breaking of specific commandments after the law was given and an accounting (ἐλλογέω) must be rendered, as distinct from sin (ἁμαρτία) before the law was given.

I am not prepared to say that Paul maintained this technical differentiation every time he used the words.  But it is a strong indication that παράπτωμα was not less than ἁμαρτία in Paul’s mind.  He continued (Romans 5:16-21 NET):

And the gift is not like the one who sinned (ἁμαρτήσαντος, another form of ἁμαρτάνω).  For judgment (κρίμα), resulting from the one transgression, led to condemnation (κατάκριμα), but the gracious gift (χάρισμα) from the many failures (παραπτωμάτων, another form of παράπτωμα) led to justification (δικαίωμα).  For if, by the transgression (παραπτώματι, a form of παράπτωμα) of the one man, death reigned through the one, how much more will those who receive the abundance of grace (χάριτος, a form of χάρις) and of the gift (δωρεᾶς, a form of δωρεά) of righteousness (δικαιοσύνης, a form of δικαιοσύνη) reign in life through the one, Jesus Christ!

Consequently, just as condemnation (κατάκριμα) for all people came through one transgression (παραπτώματος, another form of παράπτωμα), so too through the one righteous act (δικαιώματος, a form of δικαίωμα) came righteousness (δικαίωσιν, a form of δικαίωσις) leading to life for all people.  For just as through the disobedience (παρακοῆς, a form of παρακοή) of the one man many were made sinners (ἁμαρτωλοὶ, a form of ἁμαρτωλός), so also through the obedience (ὑπακοῆς, a form of ὑπακοή) of one man many will be made righteous (δίκαιοι, a form of δίκαιος).  Now the law came in so that the transgression (παράπτωμα) may increase, but where sin (ἁμαρτία) increased, grace (χάρις) multiplied all the more, so that just as sin (ἁμαρτία) reigned in death, so also grace (χάρις) will reign through righteousness (δικαιοσύνης, a form of δικαιοσύνη) to eternal life through Jesus Christ our Lord.

The law came in so that παράπτωμα may increase, but where ἁμαρτία increased, grace (χάρις) multiplied all the more, seems more like synonymous usage of παράπτωμα and ἁμαρτία rather than any kind of differentiation at all.  Similarly in 1 Corinthians 15:3 (NET) and Romans 4:25 (NET):

For I passed on to you as of first importance what I also received – that Christ died for our sins (ἁμαρτιῶν, a form of ἁμαρτία) according to the scriptures…

He was given over because of our transgressions (παραπτώματα, a form of παράπτωμα) and was raised for the sake of our justification (δικαίωσιν, a form of δικαίωσις).

It may seem at first glance that Paul meant two things in Ephesians 2:1 (NET):

And although you were dead in your² transgressions (παραπτώμασιν, a form of παράπτωμα) and sins (ἁμαρτίαις, a form of ἁμαρτία)…

But if I consider the other occurrences of παράπτωμα in Ephesians and the fact that Paul only mentioned ἁμαρτία in 2:1, it seems to me that this was a matter of completeness rather than differentiation.  In [Christ] we have redemption through his blood, the forgiveness (ἄφεσιν, a form of ἄφεσις) of our trespasses (παραπτωμάτων, a form of παράπτωμα), according to the riches³ of his grace (χάριτος, a form of χάρις).⁴  But God, being rich in mercy, because of his great love (ἀγάπην, a form of ἀγάπη) with which he loved (ἠγάπησεν, a form of ἀγαπάω) us, even though we were dead in transgressions (παραπτώμασιν, a form of παράπτωμα), made us alive together with Christ – by grace (χάριτι, a form of χάρις) you are saved!⁵

I am fairly confident that Paul did not intend sin (παράπτωμα) in Galatians 6:1 to mean something less than sin (ἁμαρτία).  But what if my argument is a straw man?  What if the real issue is between sin, whether παράπτωμα or ἁμαρτία, and such πορνεία as is not so much as named among the Gentiles?⁶  What if the former is to be dealt with in a spirit of gentleness, and the latter is to be judged, condemned and shunned?  Do not even eat with such a person.⁷

Addendum: July 29, 2019
Tables  comparing Romans 5:14; Ephesians 2:1and 1:7 in the NET and KJV follow.