Keep in mind that "schism", "the catholic church", and other relevant terms are defined differently by different individuals and groups. We have to distinguish between how Tertullian viewed himself and how his contemporaries viewed him and how he was viewed by later sources.
Eric Osborn wrote:
Meteors make few friends, so Tertullian presents a challenge to his reader. Since the Enlightenment, no ancient Christian writer has attracted more hostility….
The importance of Tertullian for cultural history is immense, and he may rightly be called the 'first theologian of the West', provided this does not limit his influence to the West or obscure his massive debt to Irenaeus….
Note the necessary qualification of G.L. Prestige, God in patristic thought (London, 1936), 97: 'He [Tertullian] was very far, indeed, from being merely the father of Latin theology. His ultimate influence on Greek theological speculation was probably very considerable.'…
He knows his philosophers better than do most Greek fathers....
It is unfortunate for this interpretation [the conclusion that Augustine wanted to apply reasoning to religious truth, whereas Tertullian didn't] that Augustine's claim that human reason has an affinity with God, quoted in support of the contrast, appeared earlier in Tertullian (an. 16.1) who seems to have provided Augustine with some of his best lines….
With one who gradually distanced himself from the rulers of the church, some evidence of change should be expected. The puzzle is one of fact: the common account of Tertullian's Montanist defection has an insecure basis. It is probable that he remained within the catholic church, despite (or because of) his allegiance to the New Prophecy. Further, his ideas on the church, its discipline and morals did not change with his increased opposition to the rulers of the empirical church which he faced. Eschatological expectation may have grown more intense; but this simply emphasized that the bride of Christ must be fit to meet her lord. There was no substantial change in his view of the church which he derived from Paul and Matthew.
Opinion concerning Tertullian's relation to the church has been divided….
It is important that Montanus and his followers did not call themselves a 'church' and that they remained loyal to the church universal….
Certainly within Tertullian's writings, the New Prophecy had no characteristics of a schism. There was no rival hierarchy as there was later with Novatianism and Donatism. While most catholics were psychics [immature Christians], not spiritual (iei. 11.1), Tertullian insisted that there were spiritual bishops (iei. 16.3) who shared his opposition to the readmission of serious sinners. Cyprian (Ep. 55.21) wrote of these earlier bishops who held such views. Cyprian would not have followed Tertullian so assiduously had Tertullian been schismatic….
This [the agreement of some bishops with Tertullian] 'is a clear demonstration that Tertullian's attacks were in the main directed more at particular bishops and their presumptuous claims than at episcopal office as such'….
Did he leave the church? It seems better to say that he did not. Yet when Cyprian took up his ideas and made the classic pattern of the catholic church, Tertullian had to appear on the outside. Tertullian never looked like a statesman; nothing came easier than the promotion of his future disqualification. Yet present disagreements were temporary; all would be set right in the coming kingdom which the paraclete was bringing….
The question [of whether Tertullian was a schismatic] is both ambiguous and obscure; but David Rankin's negative answer stands up to objections….
…Tertullian, more than Augustine, is the innovator….
His originality and openness made him an ecclesiastical failure, for the church was offended by much that he wrote, even if it could never forget him because he so clearly belonged within its life.
(Tertullian: First Theologian Of The West [New York, New York: Cambridge University Press, 2003], p. xv, p. 7, n. 20 on p. 7, pp. 31, 52, 176-177, n. 41 on p. 177, p. 251, n. 15 on p. 251, pp. 255, 257-258)
"From around 205 his [Tertullian's] writings show an increasing respect for Montanist ideas, but the style of Montanism as it was then influential in North Africa was a much-moderated form of the original Asia Minor movement, and there is no clear indication that he ever broke from the catholic community." (John McGuckin, The Westminster Handbook To Patristic Theology [Louisville, Kentucky: Westminster John Knox Press, 2004], p. 325)
"I myself have seen a certain Paul an old man of Concordia, a town of Italy, who, while he himself was a very young man had been secretary to the blessed Cyprian who was already advanced in age. He said that he himself had seen how Cyprian was accustomed never to pass a day without reading Tertullian, and that he frequently said to him, 'Give me the master,' meaning by this, Tertullian." (Jerome, Lives Of Illustrious Men, 53)