Pug Pugson had a great few answer, which I'll summarize here:

A. It all depends on what you plan on posting on these accounts...
B Both are great but people interested in your product or your customers might not be into seeing what you had for breakfast or that you got the latest iPhone...
C. So in the early days tweeting from your personal handle makes the most sense...
D. Once you grow a bit then be more active on the company account...

That pretty much answered it for me. I think I'm going to try and handle creating a second Twitter account for my product to avoid mixing content as my target customers won't want to see my personal garbage.

Just give 'peaceful piano' and 'acoustic guitar' a try - it's my morning go to for deep focus (morning = brain doesn't need high energy).

Tyler Tringas sold his incredibly successful StoreMapper and the acquisition process was incredibly simple (asset transfer), because he used all online saas products to manage and operate his business, which allowed the new owner to simple acquire accounts for different products!

You will save time, money (your time to build product is much greater in value), and stress from dealing with the issues of maintaining a supportive piece of your main product.