1.
Both sound more like manifestos than like epistles. Hebrews 13.15-25 (sometimes suspected of either being or containing an interpolation) does introduce some epistolary elements, but the rest of the epistle comes across as a treatise of some kind, as does 1 John. Also, perhaps similarly, neither epistle gives its author's name.
2.
Both start with intense, focused prologues about the son of God:
1 John 1.1-3: 1 What was from the beginning [ὃ ἦν ἀπ᾽ ἀρχῆς], what we have heard, what we have seen with our eyes, what we have looked at and touched with our hands, concerning the Word of Life — 2 and the life was manifested, and we have seen and testify and proclaim to you the eternal life, which was with the Father and was manifested to us — 3 what we have seen and heard we proclaim to you also, so that you too may have fellowship with us; and indeed our fellowship is with the Father, and with His Son Jesus Christ.
Hebrews 1.1-4: 1 God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, 2 in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world. 3 And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high, 4 having become as much better than the angels, as He has inherited a more excellent name than they.
Hebrews 1.1-4: 1 God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, 2 in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world. 3 And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high, 4 having become as much better than the angels, as He has inherited a more excellent name than they.
3.
Both epistles appear to speak about the same founding event(s):
1 John 1.1-3: 1 What was from the beginning, what we have heard, what we have seen with our eyes, what we have looked at and touched with our hands, concerning the Word of Life — 2 and the life was manifested, and we have seen and testify and proclaim to you the eternal life, which was with the Father and was manifested to us — 3 what we have seen and heard we proclaim to you also, so that you too may have fellowship with us; and indeed our fellowship is with the Father, and with His Son Jesus Christ.
Hebrews 2.1-4: 1 For this reason we must pay much closer attention to what we have heard, so that we do not drift away from it. 2 For if the word spoken through angels proved unalterable, and every transgression and disobedience received a just penalty, 3 how will we escape if we neglect so great a salvation? After it was at the first spoken through the Lord, it was confirmed to us by those who heard, 4 God also testifying with them, both by signs and wonders and by various miracles and by gifts of the Holy Spirit according to His own will.
Hebrews 2.1-4: 1 For this reason we must pay much closer attention to what we have heard, so that we do not drift away from it. 2 For if the word spoken through angels proved unalterable, and every transgression and disobedience received a just penalty, 3 how will we escape if we neglect so great a salvation? After it was at the first spoken through the Lord, it was confirmed to us by those who heard, 4 God also testifying with them, both by signs and wonders and by various miracles and by gifts of the Holy Spirit according to His own will.
But the perspectives are different. 1 John is written from the perspective of the first generation, the ones who actually experienced the founding event(s). Hebrews is written from the perspective of the second generation, the ones who heard it from those in the first generation.
4.
Both treat Jesus Christ as a mediating figure:
1 John 2.1-2: 1 My little children, I am writing these things to you that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous; 2 and He Himself is the propitiation for our sins; and not for ours only, but also for those of the whole world.
Hebrews 4.14-16: 14 Since then we have a great high priest who has passed through the heavens, Jesus the Son of God, let us hold fast our confession. 15 For we do not have a high priest who cannot sympathize with our weaknesses, but One who has been tempted in all things as we are, yet without sin. 16 Let us therefore draw near with confidence to the throne of grace, that we may receive mercy and may find grace to help in time of need.
Hebrews 4.14-16: 14 Since then we have a great high priest who has passed through the heavens, Jesus the Son of God, let us hold fast our confession. 15 For we do not have a high priest who cannot sympathize with our weaknesses, but One who has been tempted in all things as we are, yet without sin. 16 Let us therefore draw near with confidence to the throne of grace, that we may receive mercy and may find grace to help in time of need.
5.
Both stress that Jesus Christ came in the flesh:
1 John 4.1-3: 1 Beloved, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world. 2 By this you know the Spirit of God: every spirit that confesses that Jesus Christ has come in the flesh is from God; 3 and every spirit that does not confess Jesus is not from God; this is the spirit of the antichrist, of which you have heard that it is coming, and now it is already in the world.
2 John [1.]7: 7 For many deceivers have gone out into the world, those who do not acknowledge Jesus Christ as coming in the flesh. This is the deceiver and the antichrist.
Hebrews 2.14-15: 14 Therefore, since the children share in flesh and blood, He Himself likewise also partook of the same, that through death He might render powerless him who had the power of death, that is, the devil, 15 and might free those who through fear of death were subject to slavery all their lives.
Hebrews 5.7: 7 In the days of His flesh, He offered up both prayers and supplications with loud crying and tears to the One able to save Him out of death, and He was heard because of His piety.
2 John [1.]7: 7 For many deceivers have gone out into the world, those who do not acknowledge Jesus Christ as coming in the flesh. This is the deceiver and the antichrist.
Hebrews 2.14-15: 14 Therefore, since the children share in flesh and blood, He Himself likewise also partook of the same, that through death He might render powerless him who had the power of death, that is, the devil, 15 and might free those who through fear of death were subject to slavery all their lives.
Hebrews 5.7: 7 In the days of His flesh, He offered up both prayers and supplications with loud crying and tears to the One able to save Him out of death, and He was heard because of His piety.
6.
There are four verses in the NT in which Christ is called a propitiation (for sins):
Romans 3.25: 25 God displayed [Jesus Christ] publicly as a propitiation [ἱλαστήριον] in His blood through faith. This was to demonstrate His righteousness, because in the forbearance of God He passed over the sins previously committed.
1 John 2.1-2: 1 My little children, I am writing these things to you that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous; 2 and He Himself is the propitiation [ἱλασμός] for our sins; and not for ours only, but also for those of the whole world.
1 John 4.10: 10 In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation [ἱλασμὸν] for our sins.
Hebrews 2.17: 17 Therefore, He had to be made like His brethren in all things, that He might become a merciful and faithful high priest in things pertaining to God, to propitiate [ἱλάσκεσθαι] for the sins of the people.
1 John 2.1-2: 1 My little children, I am writing these things to you that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous; 2 and He Himself is the propitiation [ἱλασμός] for our sins; and not for ours only, but also for those of the whole world.
1 John 4.10: 10 In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation [ἱλασμὸν] for our sins.
Hebrews 2.17: 17 Therefore, He had to be made like His brethren in all things, that He might become a merciful and faithful high priest in things pertaining to God, to propitiate [ἱλάσκεσθαι] for the sins of the people.
The words ἱλασμός, ἱλαστήριον, and ἱλάσκεσθαι are all related, and can be used as sacrificial language, which makes sense given that Hebrews 9.11-12 (as well as many other verses) and 1 John 1.7; 5.6-8 both lay stress upon the blood of Jesus.
Interested in more connections, if you have any. Thanks.
Ben.
I think these verses are pretty theological, too. There seems to have been a "like ministers to like" principle at stake whereby one cannot help someone without being like them in important ways.