"... Mark says that the woman was ‘a Greek, born in Syro-Phoenicia’ (v.26). Since she obviously was not a Greek by nationality, ‘Greek’ is probably equivalent here to ‘Gentile’ (in distinction from Jew) or to ‘Greek-speaking’. By nationality the woman was a Syrophoenician. In those days Phoenicia belonged administratively to Syria. So Mark probably used Syrophoenician to distinguish this woman from Libyo-Phoenicians of North Africa." (Wessel, The Expositor's Bible Commentary with the New International Version - Volume 8, Matthew, Mark, Luke [Zondervan, Grand Rapids MI, 1984], p. 682; bold emphasis mine)

"... It is worthy observing, that the Holy Bible, reckoning up ... the seven nations, which were to be destroyed by the Israelites, names the Perizzites, who were not at all recited among the sons of Canaan, Gen. x., indeed, were Canaanites. See Deut. vii. 1, Josh. ix. 1, xi. 3, Judg. iii, 5, &c.

The reason of the latter (with which our business is) is to be fetched thence, that Canaan himself inhabited a peculiar part of that (northern) country, with his first-born sons, Sidon and Heth: and thence the name of Canaanites was put upon that particular progeny, distinguished from all his other sons; and that country was peculiarly called by the name of ‘Canaan,’ distinctly from all the rest of the land of Canaan. Hence Jabin, the king of Hazor, is called the ‘king of Canaan,’ Judg. iv. 2, and the kings of Tyre and Sidon, if I mistake not, are called, ‘the kings of the Hittites,’ 1 Kings x. 29.

[A Greek woman, a Syrophoenician.] Although Judea, and almost the whole world had now a long while stooped under the yoke of the Romans, yet the memory of the Syro-Grecian kingdom, and the name of the nation, was not yet vanished. And that is worthy to be noted. In the captivity, they compute the years only from the kingdom of the Greeks. They said before, ‘That the Romans, for a hundred and fourscore years, ruled over the Jews before the destruction of the Temple;’ and yet they do not compute the times to that destruction by the years of the Romans, but by the years of the Greeks. Let the Jews themselves well consider this, and the Christians with them, who reckon the Romans for the fourth monarchy in Daniel.

Therefore that woman that is here spoken of (to reduce all into a short conclusion) was a Syro-Grecian by nation, a Phoenician in respect to her habitation, and from thence called a woman of Canaan. (Lightfoot, Commentary On the New Testament from the Talmud and Hebraica [Oxford University Press, 1859; with a second printing from Hendrickson Publishers Inc., second printing 1995], Volume 2 , pp. 229-230; bold emphasis mine)

"On that day HOLY TO THE LORD will be inscribed on the bells of the horses, and the cooking pots in the LORD's house will be like the sacred bowls in front of the altar. Every pot in Jerusalem and Judah will be holy to the LORD Almighty, and all who come to sacrifice will take some of the pots and cook in them. And on that day there will no longer be a Canaanite in the house of the LORD Almighty." Zechariah 14:20-21

"Be thou ware, my son Jacob, of taking a wife from any seed of the daughters of Canaan; For all his seed is to be rooted out of the earth. For, owing to the transgression of Ham, Canaan erred, And all his seed shall be destroyed from off the earth and all the residue thereof, And none springing from him shall be saved on the day of judgment." Jubilees 22:20-21

"... Matthew uses the Old Testament place names, as he does with the word ‘Canaanite’ in v. 22, instead of Mark's ‘Syrophoenician’ (Mark 7:26, NRSV), in order to conjure up their evil connotations from bygone eras ..." (Blomberg, The New American Commentary, An Exegetical and Theological Exposition of Holy Scriptures, NIV Text, Matthew [Broadman Press, Nashville TN, 1992], p. 242; bold emphasis mine)

"Matthew describes her simply as a Chananaia, ‘Canaanite’ (the only occurrence of the word in the NT), ‘from those regions.’ The term ‘Canaanite’ has inevitable OT associations with the pagan inhabitants of Palestine displaced by the Jews and thus contrasts the woman all the more with the people of God (the term is also used for non-Jews in the rabbinic literature; cf. m. Qidd. 1:3; b. Sota 35a). As Mark's description of her as a ‘Ellenis, ‘Greek’, suggests, she was Hellenized to some extent, and almost certainly the conversation between her and Jesus would have been held in Greek.’ (Hagner, World Biblical Commentary, Matthew 14-28 [Word Books, Publisher, Dallas TX], p. 441; bold emphasis mine)

"As Jesus went on from there, he saw a man named Matthew sitting at the tax collector's booth. ‘Follow me,’ he told him, and Matthew got up and followed him. While Jesus was having dinner at Matthew's house, many tax collectors and ‘sinners’ came and ate with him and his disciples. When the Pharisees saw this, they asked his disciples, ‘Why does your teacher eat with tax collectors and "sinners"?’ On hearing this, Jesus said, ‘It is not the healthy who need a doctor, but the sick. But go and learn what this means: "I desire mercy, not sacrifice." For I have not come to call the righteous, but sinners.’" Matthew 9:9-13

There is no conflict between the accounts. The explanation has to do with the fact that the two writers - Matthew and Mark - are directing their respective documents to different segments of that ancient society. Thus, they adapt their terminology to understanding of their targeted recipients.

1. Matthew tailors his record for the Jews. This is apparent from a number of different vantage points. For example, his heavy reliance upon the Old Testament scriptures indicates this. He is writing for those who accept the Old Testament Scriptures as authoritative. Accordingly, with reference to this woman who lived in the sea-coast region in northwestern Palestine, he calls her a "Canaanite" lady. The pagan inhabitants of the land which Israel conquered under Joshua were known as Canaanites, being descended from Canaan, the grandson of Noah (Gen. 9:18). Many of the Canaanites had been pushed northward into Phoenicia when the Hebrews invaded the territory. This dear woman was designated as a Canaanite because her ancestry was of these despised enemies of Israel.

2. Mark, on the other hand, is writing for the benefit of the Romans, who controlled the Mediterranean world of the first century. His Roman interest is seen, for instance, in the Latin forms which he employs to render Greek equivalents (cf. 3:17; 5:41; 7:11, 34; 14:36; 15:22, 34). This woman lived in Phoenicia (which, politically speaking, belonged to the province of Syria). Hence, she is designated a Syrophoenician. She is further denominated as a "Greek" because she had absorbed the Greek culture, obviously speaking that language. In the New Testament, the term Hellen ("Greek") frequently is used in the generic sense of simply a "Gentile" (Jn. 7:35; Acts 9:29; Rom. 1:16, etc., see: F.W. Danker, Greek-English Lexicon, Chicago: University of Chicago, 2000, 318).

These two texts regarding this noble Gentile woman do not conflict at all. Rather, they are wonderfully complimentary, reflecting the individuality of the two sacred writers. (Source)