As to the meaning of the word tradition, that seems to be a major difference, so:
Hold the traditions (krateite tav paradoseiv). Present imperative of kratew, old verb, to have masterful grip on a thing, either with genitive (Mar_1:31) or usually the accusative as here. Paradosiv (tradition) is an old word for what is handed over to one. Dibelius thinks that Paul reveals his Jewish training in the use of this word (Gal_1:14), but the word is a perfectly legitimate one for teaching whether oral, by word (dia logou), or written, by epistle of ours (di epistolhv hmwn). Paul draws here no distinction between oral tradition and written tradition as was done later. The worth of the tradition lies not in the form but in the source and the quality of the content. Paul in 1Co_11:23 says: "I received from the Lord what I also handed over (paredwka) unto you." He praises them because ye "hold fast the traditions even as I delivered them unto you." (A.T. Robertson)
The tradition which he taught them was the revelation that he received from the Lord, and which is now recorded in Scripture.
traditions--truths delivered and transmitted orally, or in writing (2Th_3:6; 1Co_11:2; Greek, "traditions"). The Greek verb from which the noun comes, is used by Paul in 1Co_11:23; 1Co_15:3. From the three passages in which "tradition" is used in a good sense, Rome has argued for her accumulation of uninspired traditions, virtually overriding God's Word, while put forward as of co- ordinate authority with it. She forgets the ten passages (Mat_15:2-3, Mat_15:6; Mar_7:3, Mar_7:5, Mar_7:8-9, Mar_7:13; Gal_1:14; Col_2:8) stigmatizing man's uninspired traditions. Not even the apostles' sayings were all inspired (for example, Peter's dissimulation, Gal_2:11-14), but only when they claimed to be so, as in their words afterwards embodied in their canonical writings. Oral inspiration was necessary in their case, until the canon of the written Word should be complete; they proved their possession of inspiration by miracles wrought in support of the new revelation, which revelation, moreover, accorded with the existing Old Testament revelation; an additional test needed besides miracles (compare Deu_13:1-6; Act_17:11). When the canon was complete, the infallibility of the living men was transferred to the written Word, now the sole unerring guide, interpreted by the Holy Spirit. Little else has come down to us by the most ancient and universal tradition save this, the all-sufficiency of Scripture for salvation. Therefore, by tradition, we are constrained to cast off all tradition not contained in, or not provable by, Scripture. The Fathers are valuable witnesses to historical facts, which give force to the intimations of Scripture: such as the Christian Lord's day, baptism, and the genuineness of the canon of Scripture. Tradition (in the sense of human testimony) cannot establish a doctrine, but can authenticate a fact, such as the facts just mentioned. Inspired tradition, in Paul's sense, is not a supplementary oral tradition completing our written Word, but it is identical with the written Word now complete; then the latter not being complete, the tradition was necessarily in part oral, in part written, and continued so until, the latter being complete before the death of St. John, the last apostle, the former was no longer needed. Scripture is, according to Paul, the complete and sufficient rule in all that appertains to making "the man of God perfect, throughly furnished unto all good works" (2Ti_3:16-17). It is by leaving Paul's God-inspired tradition for human traditions that Rome has become the forerunner and parent of the Antichrist. It is striking that, from this very chapter denouncing Antichrist, she should draw an argument for her "traditions" by which she fosters anti-Christianity. Because the apostles' oral word was as trustworthy as their written word, it by no means follows that the oral word of those not apostles is as trustworthy as the written word of those who were apostles or inspired evangelists. No tradition of the apostles except their written word can be proved genuine on satisfactory evidence. We are no more bound to accept implicitly the Fathers' interpretations of Scripture, because we accept the Scripture canon on their testimony, than we are bound to accept the Jews' interpretation of the Old Testament, because we accept the Old Testament canon on their testimony. (Jamieson, Faucett, and Brown)
"And hold the traditions which ye have been taught." On the word traditions, Cmt. on Mt 15:2. It means properly things delivered over from one to another; then anything orally delivered--any precept, doctrine, or law. It is frequently employed to denote that which is not written, as contradistinguished from that which is written, (comp. Mt 15:2,) but not necessarily or always; for here the Apostle speaks of the "traditions which they had been taught by his epistle." Comp. Cmt. on 1Co 11:2. Here it means the doctrines or precepts which they had received from the apostle, whether when he was with them, or after he left them; whether communicated by preaching or by letter. This passage can furnish no authority for holding the "traditions" which have come down from ancient times, and which profess to have been derived from the apostles; for
(1.) there is no evidence that any of those traditions were given by the apostles;
(2.) many of them are manifestly so trifling, false, and contrary to the writings of the apostles, that they could not have been delivered by them;
(3.) if any of them are genuine, it is impossible to separate them from those which are false,
(4.) we have all that is necessary for salvation in the written word; and
(5.) there is not the least evidence that the apostle here meant to refer to any such thing. He speaks only of what had been delivered to them by himself, whether orally or by letter; not of what was delivered from one to another as from him. There is no intimation here that they were to hold anything as from him which they had not received directly from him, either by his own instructions personally or by letter. With what propriety, then, can this passage be adduced to prove that we are to hold the traditions which professedly come to us through a great number of intermediate persons? Nowhere is the evidence here that the church was to hold those unwritten traditions, and transmit them to future times?
"Whether by word." By preaching, when we were with you. It does not mean that he had sent any oral message to them by a third person.
"Or our epistle." The former letter which he had written to them.
{*} "Traditions" "doctrines" (Albert Barnes)
Here are some other authorities for you to ponder over.
DHK