How do you use ' ともうします'?
1. What scenarios would be appropriate to use 'コーネレイアスともうします。' over '私の名前はコーネリアスです。' ?

2. And asking the converse: are there inappropriate times to use it ' ともうします' ?

(Sorry I don't really know how to ask these questions in Japanese yet.)

Question

How do you use ' ともうします'?
1. What scenarios would be appropriate to use 'コーネレイアス ともうします。'  over  '私の名前は コーネリアスです。' ?

2. And asking the converse: are there inappropriate times to use it ' ともうします' ?

(Sorry I don't really know how to ask these questions in Japanese yet.)

Answer

～と申します。(もうします。) 
:formal Greetings and Introductions
ex)business/ business email/job interview

私は〇〇です。
:Formal&Informal Greetings and introductions

Have a good day!