Better is a very subjective term, but I'll try to lay down the pros and cons for both approaches.
WordPress - Pros
- The ecosystem. WordPress is used by 27% of the web, and the amount of support you can get is amazing. Woocommerce, which is the leading ecommerce plugin for WordPress, is used on almost 39% of ecommerce sites.
- Extensibility. The software is robust, and extensible. For anything you want to do, there is a plugin. If there's not one for your particular needs, you can build one.
- Open Source. Being open source is one of the biggest plus for WordPress. This basically implies that the source code is always going to be there, and one single company/developer going out of business will have no impact on the ecosystem.
- Framework. The WordPress core is a very robust, and tested framework for any kind of website that you might want to build. It gives you custom data types, a full-featured user management system, a full featured event-driven hooks system, and a full-featured templating engine. In addition, it has a REST API if you're going to build mobile apps in the future.
- Community. Self-explanatory.
WordPress - Cons
- Ecosystem. It is very easy to fall in the hands of the wrong agency/developer.
- Extensibility. You may fall into a rabbit hole with plugins. You need to keep this in check proactively to make sure your site runs smooth.
- Monolithic, and not completely Object-oriented.
Custom website - Pros
- Everything will be according to the specification.
- You can probably build whatever you want, and choose whichever language you want(even C++, Python, Go)
- It's probably easier to build full-fledged web apps this way.
Custom website - Cons
- No ecosystem. If your developer/agency runs away, or goes out of business, you're doomed.
- Vendor lock-in. If you hire an external agency, you're pretty much locked in with them.
- No regard for content portability. In most of the cases, custom website developers completely disregard data portability. Building this means more money from your pocket.
- Security. WordPress, being open source, has a lot of people looking at the code to make sure it works okay. Your custom application WILL have undiscovered bugs, and there's very little chance you'll be able to discover them before and attacker does.
- Extensibility. Self-explanatory.
In a nutshell, if WordPress is good enough for Time, NY Times, CNN, Fortune.com, Microsoft Studios, TED, Techcrunch, NBC, Skype, etc etc, it's probably good enough for you. What you build is completely up to the developer/agency that you choose, and the amount of resources you want to put in it.
Disclaimer: I work for Automattic Inc., but the views expressed here are completely my own, and I'm not representing my employer here. All of these views have been a result of my 8 year career in Web development and working with local and international clients/employers.