The gemara has a variety of rulings that we apply to situation where 2 people are fighting over a particular object. When the object is being held by Reuven and Shimon is claiming that it belongs to him, we apply the standard rule of Hamotzi l'chaveiro alav ha'raya (as rashi points out). In a situation where they are both holding on to the object such as our mishnah, we view it as if each is muchzak in half the talis so that we apply the same rule - each one keeps what he is holding, therefore we divide (as tosafos d.h. v'yachloku and the Rosh explain). A shavuah shouldn't even be necessary, but it's purpose is to prevent theives from grabbing their friends clothes and make fraudulent claims (gemara 3a). However, if they are arguing about something that is not in the possession of either one of them, such as the case of a boat where each one is claiming that it belongs to him, Tosafos explains that since there is no muchzak we apply the rule of כל דאלים גבר, meaning that the "stronger one" wins. The Rosh elaborates that this is really a technique that beis din uses to stay out of the situation. Although beis din is required to involve themselves when they see someone taking from someone else, in this case since no one person is a muchzak they are not required to get involved (the rosh also offers some justification as to why this method would likely result in the rightful owner winning).
The mishna considers the two who are holding on to it to both be muchzak in the talis. But, in the second case in the mishna, they are in exactly the same positions physically grabbing on to it, yet we say that Reuven gets 3/4 and Shimon gets 1/4. Why don't we disregard their claims and view their status as being each muchzak in half to split it evenly as we do in the first case? It seems that the status of being muchzak is not a result of their actions of holding it alone, rather it is a combination of their holding it in conjuction with their claims. Meaning, only because they are both claiming it to be fully theirs do we consider them each to be muchzak on half. Therefore, in the second case where both agree that half belongs to Reuven, Reuven is automatically considered a muchzak on 3/4 and Shimon a muchzak on 1/4.
Rashi (2a) and Tosafos (2b) argue what the din will be in a case where 2 people are holding a talis, each claiming that it belongs to him because he wove it himself (unlike the mishna where they are claiming to have found it or purchased it). The difference between this case and the mishna case is that in the case where they are claiming to have woven it, one of them is definitely not telling the truth (whereas in the mishna it is possible that they picked it up together). Rashi holds that in the case where they claim to have woven it, since one of them is definitely a thief, we apply the din of יהא מונח עד שיבא אליהו. However, Tosafos says that we apply the same din as our mishna of dividing with a shavua (the fact that one is a thief will make the shavua even more effective because even a thief is not suspect to swear falsely). The Rosh proves that Tosafos is correct from the gemara 3a which asks why the by מנה שלישי do we say יהא מונח rather than יחלוקו? The gemara says that since the third 100 zuz definitely belongs to either Reuven or Shimon, but definitely doesn't belong to both, we can't divide it. Rather than using the rationale that we don't divide since one of them is definitely cheating (as the gemara says by r' yossi), the gemara implies that even if one of them is definitely cheating we will still divide so long as it is still possible that it belongs to both of them. Therefore, in the case where each one claims to have woven it, where one of them is definitely cheating, we would still divide, since it is possible that they both own it.