I'll give my biased response, but I hope you can see I'm trying to be fair and honest too. Note I'm one of the founders of HashiCorp and original creators of Vault.
If your challenge is just storing static secrets (think key/value), Vault is probably overkill for you today. Secrets Manager will work fine, or even a smaller KMS-based solution or something. We're working on making Vault a LOT easier to get started with so this probably won't be true for long, but its probably true today. But it is important to understand the tradeoffs of making these decisions.
The value of Vault is in the fact that it does so much more: dynamic secrets, automatic rotation, certificate management, encryption-as-a-service, etc. And that it integrates with so many systems: log in with AWS IAM, or K8S service principles, or OIDC (Google, GitHub, etc.). And it has a single policy and auditing system to back all this.
Usually Vault becomes VERY beneficial when you're juggling multiple "secret-like" solutions: diff password solution from key management from PKI etc etc OR you want to adopt more modern practices like dynamic credentials OR you want a way to centrally govern secret-like things.
Vault literally scales from solving the needs of a small team (static KV) to being used by some of the Fortune 10 to back their entire corporate secret, PKI, encryption, signing requirements in a centralized way. I think that's kind of neat.