Open Source typically refers to software that has code released under some type of free software license (GPL, BSD, Apache, etc.). Users and developers have access to the source code and can modify and redistribute it. Proprietary software usually means the source code is closed and/or the EULA (end user license agreement) forbids redistribution, copying, etc.
The benefit of open source software is that there is a large community involved in supporting it, finding bugs, and making it secure. For a server, this means that your software is not going anywhere even if the hardware vendor goes out of business. With proprietary software, you are at the mercy of the vendor or software developer when you need something fixed or have security problems.
The benefit of proprietary software is usually only evident if you make web applications in-house and do not intend to release that software to the outside public. In such cases you may view the software as your intellectual property. The cost for maintaining each varies depending on the setup and who you have maintaining it. If you or your employees have the knowledge to maintain free software, you can save a considerable amount of money.