In terms of architecture, it's entirely different. Rather than being a centralized service (like mattermost), Matrix is a decentralized protocol that can be fed

In terms of architecture, it's entirely different. Rather than being a centralized service (like mattermost), Matrix is a decentralized protocol that can be federated across servers. And vector.im is just one (web) interface for using this protocol. It just happens that it can be used for similar things that Slack can. It's more like IRC (but distributed/federated)