The main benefit of Multipath TCP when used in multihoming scenarios compared to BGP-based multihoming is that Multipath TCP capable hosts can use different paths simultaneously while BGP-based multihoming would provide one path for each client server pair.
With Multipath TCP, multihoming must be exposed to the server. For example, consider a small enterprise network connected to two different providers, A and B. With BGP-based multihoming, you assign address C to your server and advertise it through the two providers via BGP. BGP decides which path will be used and the ASPath metric used by BGP is far from being the most accurate metric to evaluate the quality of a path.
With Multipath TCP, you should ask a block of addresses from both A and B and assign one address from each provider you your server, say A.1 and B.1. Both addresses are advertised in the DNS. Address A.1 is always reachable via provider A and B.1 always reachable via provider B. When a TCP connection reaches the server, say over A.1, it will also advertise address B.1 using Multipath TCP and a second subflow will be established. Multipath TCP will then regulate the usage of the two paths in function of the amount of congestion on each path. If one path fails, Multipath TCP will perform failover automatically.