According to my Mantis Bible, the English word "boasts" is the Hebrew word "halal"; a prim. root; to be boastful, to praise: - It is most often translated praised or praise. It is also found in Gen. 12:15; Judges 16:24; I Sam. 21:13; 2 Sam. 14:25; 22:4; I Ki. 20:11; I Chr. 16:4, 10; 16:25, 36; 23:5, 30; 25:3; 29:13; 2 Chr. 5:13; 7:6; 8:14; 20:19, 21; 23:12, 13; 29:30; 30:21; 31:2; Ezra 3:10-11; New. 5:13; 12:24; Job 12:17; Ps. 5:5; 10:3; 18:3; 22:22-23, 26; 34:2; 35:18; 44:8; 48:1; 49:6; 52:1; 56:4, 10; 63:5, 11; 64:10; 69:30, 34; 73:3; 74:21; 75:4; 78:63; 84:4; 96:4; 97:7; 102:8, 18; 104:35; 105:3, 45; 106:1, 5, 48; 107:32; 109:30; 111:1; 112:1; 113:1, 3, 9; 115:17, 18; 116:19, 117:1, 2; 119:164, 175; 135:1, 3, 21; 145:2, 3; 146:1, 2, 10; 147:1, 12, 20; 148:1, 2, 3, 4, 5, 7, 13, 14; 149:1, 3, 9; 150:1, 2, 3, 4, 5, 6; Prog. 12:8; 20:14; 25:14; 27:1, 2; 28:4; 31:28, 30, 31; Ecc. 2:2; 7:7; Song 6:9; Isa 38:18; 41:16; 44:25; 45:25; 62:9; 64:11; Jer. 4:2; 9:23, 24; 20:13; 25:16; 31:7, 46:9, 49:4; 50:38; 51:7; Exek. 26:17; Joel 2:26; and Nah. 2:4.
The English word translated "knows" is the Hebrew word "yada" a prim. root; to know. It refers to ability, acknowledge, acquaintances, acquainted, aware, becomes known, bring forth, cared, chosen, clearly understand, cohabit, comprehend, concern, concerned, consider...It is most often translated know (542 times) Known (65 times)...other less common translations are raped, satisfied, understand, provided. There are too many places this word is found in the Bible to identify each.
Jer. 9:23-24 is quoted in I Cor. 1:31 regarding boasting in the LORD. We also see this concept in Ps. 20:7; 44:8; 2 Cor. 10:17; Gal. 6:14. Instead of boasting in what we have or have done, our boast need be in Christ who gave us the strength and power to achieve what we have achieved.
Regarding "know", there is a huge difference between knowing about someone and truly "knowing" them. God wants us to truly "know" Him.
I hope this is helpful.
Gloria