I think the real question you are asking here is Vault's Integrated Storage (aka Raft) vs. Consul (external storage).
There are several aspects to this but the top 3 are:
REDUCED COST & REDUCED OPERATIONAL OVERHEAD & LACK OF CONSUL EXPERTISE ON YOUR
If you want to reduce operational costs (by reducing administrative overhead), then choose Integrated Storage
(Raft) is the preferred choice. In a standard cluster configuration,
you only need 5 Vault nodes, which translates to VMs on AWS, Azure,
GCP, etc. With Consul you will need 3 Vault VMs + 5+ Consul VMs, so
a minimum of 8 - see Reference Architecture with Consul
INSPECTING VAULT DATA
If you need to inspect Vault's data frequently, then Consul as an external storage is the better option
SIDECAR + SERVICE DISCOVERY
If you rely on Consul's service discover and sidecar proxy pattern, then you need it. By contrast, if you are only leveraging Vault's secret management features and capabilities, then the Integrated Storage (Raft) would do just fine.
Take a look at the checklist at the end of this article and at the reference architectures for more clarity.
For leveraging different type of backends with Vault, you can take a look at this Pluralsight course: Managing Access and Secrets in HashiCorp Vault
There courses covering Consul as well, of course. But generally, Consul is a a lot more than just a Vault backend for storing data.