Founder of WOW Branding, co-author of Logy Savvy. Defining core values, core purpose, positioning and brand messaging
The main advantage of having the company and the product/service sharing the same name is that it is much more cost effective to build the brand in the early stages. You also need to consider what relationship any future products are going to have with your first (if any) - do they complement, compete, same markets/customers, etc.
Generally, you will be better off by keeping the names the same. Think about how you pitch your company vs the product - is it a different story? Which name do you want people to remember? Think about where the names would live - business cards, urls, websites, app (icon), signage, etc.
There are countless successful examples of different brand naming structures that work - there is no "best" way. Keep it simple.
We wrote a book on naming and identity design a few years back. Happy to send you the first chapter pdf to see if it can help.
Dann Ilicic WOW Branding