I'm the only developer working on an open source project (MPL v2.0 license, not sure if it does matter) and I have a single-person LLC I use for some consulting works.
What are the pros and cons of copyrighting the code under my own name vs the name of the company?
Given that you are the sole owner of your LLC, and depending on how liberal the open source license is, the difference isn't meaningful.
Tax
If the code is developed for the LLC as a work-for-hire, that is your LLC pays yourself, it may be classed as an R&D expense, which may be advantageous to do. Consult your local tax laws or your accountant. If you write the code yourself, your LLC will just be using a free resource, which has no tax implications.
Business ownership transfer
If you ever sell your LLC, or it goes bankrupt, and the LLC is the copyright holder, the code is considered the LLC's property and will be transferred along with it, which effectively means you lose ownership of your code.
Rights
There are certain rights that copyright holders have that typically aren't granted by open source licenses, like relicensing, or closing the project later. If you ever need to do something not allowed by the license, technically one of the entities (you, your LLC) needs to give permission to the other. It doesn't really matter unless you sue yourself, say for insurance purposes.
Expiration
The copyright expiration is different depending on whether the author is human or not (anonymous, pseudonymous, work-for-hire). Typically, and for the US, it's author's life + 70 years, or 95 years from publication or 120 years from creation whichever is shorter. Given that these terms are stupendously long the difference isn't meaningful.
