From Anstey’s Doctrinal Definitions:
This word signifies a new beginning in a believer’s life—a new outward moral change, resulting from being saved and sealed with the Holy Spirit. This change is something that people should see in the life of every believer. W. Scott said, "The washing of regeneration can be discerned by the eye of man, as it is an outward change" (Doctrinal Summaries, p. 28).
The “washing of regeneration” (Titus 3:5) refers to the moral cleansing of a believer’s life, resulting from his separation from his old pre-conversion lifestyle according to the flesh. It has to do with a person’s passage from the old order of life in which he once lived into the new order of life that is consistent with Christianity. Hence comes the often-heard comment, "He has cleaned up his life; he’s on the straight and narrow now!" If this change is a real work of God in a person’s life, it is a result of the washing of regeneration. The "washing" involved in regeneration is connected with that which the waters of baptism signify for the baptized. The person is thereby set in a new clean place on earth in the house of God, wherein he is to walk in holiness as a Christian (Psa. 93:5; Heb. 12:14; 1 Thess. 4:7; 1 Peter 1:16). Paul emphasized to Titus the need for regeneration among the believers on the island of Crete because there was a clear lack of practical righteousness in their lives (Titus 1:12).
Notwithstanding, Christians from all schools of thought generally do not understand regeneration. They think that since the word "regenerate" means to re-start something, it is referring to being born again, and thus, use these terms interchangeably. However, while new birth and regeneration both refer to a new beginning, they are two different beginnings in a person’s life.
New birth (which occurs first in a person’s history) is an inward new beginning in the soul through receiving a new life from God. An evidence of this will be seen in his seeking after God.
Regeneration is an outward new beginning of a believer’s life, consequent upon his being saved and sealed with the Holy Spirit. The evidence of this will be seen in his breaking off of unholy associations and habits, and his adopting the Christian lifestyle.
Mr. Darby said, “Regeneration is not the same word as ‘being born again,’ nor is it used so in Scripture” (J. N. Darby Translation footnote on Titus 3:5). W. Scott said, “New birth is not the same as regeneration, which the latter term only occurs twice in the New Testament (Titus 3:5; Matt. 18:28). The former term refers to an inward work; the latter to an outward change” (The Young Christian, vol. 2, p. 131). He also said, “It [regeneration] is almost universally regarded as equivalent to new birth, but it is not so in Scripture. Regeneration is an objective state or condition, while the new birth is the expression of an inward and subjective state” (Bible Handbook, Old Testament, p. 372).
New birth and regeneration both involve a bath, which signifies washing or cleansing. The “bath” involved in new birth signifies an inward washing in the soul through receiving a new clean life from God (John 13:10 – W. Kelly Trans.; 1 Cor. 6:11), whereas the “bath” in regeneration signifies an outward washing of the person's life in a practical sense (Titus 3:5 – J. N. Darby Trans. footnote). The former is effected by the “quickening” of the Holy Spirit (John 6:63); the latter is effected by the “renewing” of the Holy Spirit (Titus 3:5). The practical effect of regeneration is illustrated in a new believer at his baptism. He leaves his belongings (his cigarettes, flask of liquor, worldly magazines, etc.) at the water's edge, and wades into the water. After being baptized and coming out of the water, he goes on his way with the Christians who attended the baptism. But someone calls to him, "John, you forgot your things." He replies, "Leave them there; they belong to the old John." This should be the practical effect of the washing mentioned in regeneration; there is a separation (and thus a cleansing) from the old lifestyle. It signifies a new beginning in a believer's life that should be observable by all.
To further complicate the misunderstandings that rise from assuming that regeneration is the same as new birth, many Christians have the idea that regeneration is a miraculous work of God in renewing or re-making a person's nature when he believes the gospel. They think that it is an infusing of new life into a person’s old nature, making it anew. Based on this mistaken idea, most Reformed theologians, and many evangelical preachers as well, teach that Christians do not have two natures, but rather, one nature that has been regenerated! This makes new birth and regeneration to be nothing more than a rehabilitation of the flesh. The truth is, that the old nature is not capable of being improved and thus it cannot be rehabilitated. Scripture says, "It is not subject to the law of God, neither indeed can be" (Rom. 8:7). The flesh simply cannot be made subject to God. The Lord taught this to Nicodemus. He said, "That which is born of the flesh is flesh" (John 3:6). That is, men could apply many devices to the flesh in an effort to improve it—the influence of culture, education, religion, etc.—but the end result is that nothing changes morally. The flesh is still the flesh. And this is why the Lord said to him, "Ye must be born again." That is, men need an altogether new life and nature that is separate and distinct from the old nature imparted to them (John 3:7).
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Matthew 19:28 also speaks of “regeneration” but in a slightly different way. It is referring to the new, outward, moral order of life that will pervade the world to come—the Millennium (Psa. 72:8; Zech. 14:9). Men in that day will be forced to live according to righteous principles for fear of death (Isa. 32:1; Psa. 101:7-8; Zech. 5:1-4).