Flarum

vs

bbPress

Create an online community that your customers will love. Vanilla's forum software is used by top brands to engage customers, drive loyalty and reduce support costs. The software is provided free and open-source as a self-hosted solution and can be bought by businesses and enterprises as a managed and hosted solution.

One of the world most popular, free, open-source and self-hosted forum software, written in PHP. With phpBB you can create communities that can be hosted with most hosting providers. The software package is built and designed from the ground up to to be extensible and provides customization option through themes and feature extensions through plugins. The extensive database of user-created extensions and styles database, easy of use, and the simple installation process has made phpBB a common choice among self-hosted forums and communities.

Discourse is an open source internet forum and mailing list software. The application is written with Ember.js and Ruby on Rails, and can be self-hosted in the cloud or on local hardware for free. Discourse also provides managed hosting starting at $100/month, so that you can focus on building a community rather than installing software and managing servers.

Disqus is a Comment-as-a-Service provider that makes it easy to add comments to any web property. It's one of the most popular comment plugins on the market today. With Disqus, publisher can bring their content to life, fuel colorful discussions, efficiently moderate comments, and build engaged communities. The plugin takes only a matter of minutes to setup and is as easy as installing a WordPress plugin or adding a code snippet to your website.

Simple Machines Forum or SMF in short is a free and open-source forum software. SMF can be used to build your own online community that you self-host on a host of your choice. Simple Machines Forum comes packed with features for customization, extension and management of your community. The template engine puts you in control of the design of your community. The plugin architecture makes it easy to extend and provide further feautres though third-party built packages. 

vBulletin is a powerful commercial forum software written in PHP. The software is built to be easily extended through plguins and customized with themes. vBulletin is used to power many of the largest forum, communities and social site across the web. Over 100,000 sites are currenty powered by vBulletin.