We would not recommend it unless you have to move to Azure for billing purposes.
There is no difference in the features available or plan benefits as everything is still managed through the Raygun platform (except billing). We also don't support Pulse in Azure as yet so it then becomes harder to add this to your plan if you want it in the future and you'll end up with separate billing for the two products.
The other benefit is you will get a much better customer support experience when it comes to billing and account management through us directly than you would with Azure.
However, if you do decide to move then the easiest may to migrate is to sign up with Azure and create a new plan. You can then associate this to your existing user account and migrate your apps from one plan to the other. Once done and you have reconfigured your users and teams, then you can cancel the old plan.