Both companies are already competing because they keep the two most important citation and impact databases in their portfolios with Scopus (owned by Elsevier) and the Web of Science (previously owned by Thomson Scientific but since 2017 by Clarivate Analytics). The only competitor in this field is Google Scholar. For all those who do not know publons, there is some information in Telepolis [in German language]. In short: Publons wants to reward reviewers, if only symbolically, by receiving diligence points (or credits) for reviews. Per se this is not a bad idea, because reviewing submissions to scientific journals is mostly neither symbolically nor financially rewarded.
Among other features publons also has an interface to the Altmetrics service Impactstory (incidentally, a competitor to PLUM Analytics), so the credits a scientist gained for reviews via publons are displayed in his impactstory profile.
Graduate sociologist, information scientist (PhD degree), working for the Saarland University and State Library (Germany), owner of the consulting agency scinoptica, journalist. ORCID: 0000-0002-3500-3119