Both βούλομαι and θέλω, which latter word is of much more frequent use, are translated ‘to will' and the latter ‘to desire,' with other variations of both in the A. V.
The distinction of the two words seems to be justly designated (at least as to their N. T. use) thus: θέλω, expresses in general ‘to exercise the will’ ― the will which proceeds from inclination or desire. βούλομαι is the will which follows from deliberation, and involves a carefully-weighed purpose.
They occur together in Matt. 1:19: Joseph not desiring (θέ.) to expose Mary publicly, purposed (βο) to have put her away secretly. In 1 Tim. 5:11, as to the younger widows not being put on the list, they will (θέ.), their desire is, to marry: in verse 14 Paul's deliberate judgment and will is that they should; "I will (βο.) therefore."
In Philem. 1:13 the apostle "would have" (βο.), was desirous of, keeping Onesimus with him, but without Philemon's mind "willed (θέ.) to do nothing." Also in 1 Tim. 2:4, as to "God our Savior, who desires that all men should be saved," it is θέλω: see Eze. 18:23 (LXX, Vat.) "Have I any pleasure at all that the wicked should die?" (with which may he compared 1 Cor. 12:18 "as it hath pleased [θέ.] him," and 15:38): in 1 Tim. 2:8 it is βούλομαι as in chap. 5:14 "I will therefore," the active wish being implied.
For θέλω see more fully John 6:67, "Will ye also go away?" not simply the act, but the will to do it: ‘is it your will or disposition?’ ‘are ye also disposed to go away?' Luke 15:28: "he was angry, and ‘would' not go in." Mark 6:19: "would have killed him." 1 Cor. 10:27: "and ye be ‘disposed' to go." Matt. 17:12: "whatsoever they listed." Compare the use of θέλω, for ‘I would' and ‘would not' in Rom. 7:15, 16, 18, 19, 20, 21, in opposition to ‘I hate,' ‘I do.' Also 1 Cor. 7:7: "I would that all men were even as myself." Rom. 9:16: "it is not of him that willeth." Mark 9:35; 12:38, may be added where it is ‘desire' and ‘love.'
For βούλομαι, see Matt. 11:27: "he to whomsoever the Son will (or ‘wills to') reveal him." Heb. 6:17: "wherein God, willing more abundantly to show unto the heirs of promise the immutability of his counsel." (The word ‘counsel' is the substantive βουλή: it occurs ten times so rendered in the A. V.) So 1 Cor. 12:11 of the "Spirit, dividing to every one severally as he will." Jas. 1:18: "of his own will" — ‘having so purposed or willed:' it was the fruit of His own mind, and so a free gift (ver. 17). 2 Pet. 3:9: "not willing that any should perish, but that all should come to repentance," thus becomes clear as the expression of His purpose. Verse 5 may be referred to for the force of θέλω: "this they are willingly ignorant of," or "this is hidden from them through their own willfulness."
With the above may be compared the judicial will' or would' (βούλομαι) of Acts 18:15; 22:30; 23:28 25:22; 28:18; and Pilate's use of it in putting the momentous decision to the Jews, John 18:39. In 1 Tim. 6:9, it will be seen that "will be rich" (βο.) includes the idea of purpose; as also Jas. 4:4, with regard to the friendship of the world.
Eph. 1:11 brings together the substantives formed from the two words: "after the counsel (βουλή) of his own will (θέλημα)." The latter is used in chap. 2:3: "the ‘desires' of the flesh and of the mind;" and the consistent distinction of the former is maintained in 1 Cor. 4:5, "and will make manifest the counsels of the hearts."
βούημα only occurs in Acts 27:43, translated ‘purpose,' and Rom. 9:19, ‘will.'
πρόθεσις is another word connected with ‘purpose,' which is its fitting translation: it is, according to the verb προτίθημι, what I set before myself, and so propose and determine. See for the verb Rom. 1:13 and Eph. 1:9. The substantive, πρίθεσις, is found in connection with the ‘shewbread' in its physical force of setting forth, or placing a thing in view, Heb. 9:2 ― the O. T. expression for the rite of the "setting forth of the loaves"; but, following the verb, it is "purpose of heart" in Acts 11:23 (comp. 2 Tim. 3:10), and in five other passages applied to the purposes of God's heart, Rom. 8:28; 9:11; Eph. 1:11 (where it is closely associated with βουλή) and θέλημα); 3:11; 2 Tim. 1:9.