Be angry and sin not=don't sin by being angry.
"Wrath" is something forbidden to us if we are being obedient.
Gal 5:19 Now the works of the flesh are manifest, which are these; Adultery, fornication, uncleanness, lasciviousness,
Gal 5:20 Idolatry, witchcraft, hatred, variance, emulations, wrath, strife, seditions, heresies,
Gal 5:21 Envyings, murders, drunkenness, revellings, and such like: of the which I tell you before, as I have also told you in time past, that they which do such things shall not inherit the kingdom of God.
Eph 4:31 Let all bitterness, and wrath, and anger, and clamour, and evil speaking, be put away from you, with all malice:
Eph 4:32 And be ye kind one to another, tenderhearted, forgiving one another, even as God for Christ's sake hath forgiven you.
Col 3:8 But now ye also put off all these; anger, wrath, malice, blasphemy, filthy communication out of your mouth.
Col 3:9 Lie not one to another, seeing that ye have put off the old man with his deeds;
1Ti_2:8 I will therefore that men pray every where, lifting up holy hands, without wrath and doubting.
And this one is GREAT!
Jas_1:19 Wherefore, my beloved brethren, let every man be swift to hear, slow to speak, slow to wrath:
Jas_1:20 For the wrath of man worketh not the righteousness of God.
Get that? The "wrath of man" worketh NOT the righteousness of God. There's your 'difference'.
"Vengeance" is "MINE" sayeth the Lord. I (the LORD) will repay. Of course, that doesn't mean there aren't plenty of people who make themselves out to be god by taking on (poorly imitating) His attributes. Impossible to be 'angry' at a brother while still taking the beam out of our own eye.