They are probably both looking over their shoulders at each other (and others, like AWS DevOps), and copying the features most reported as missing, to the point that it's going to be difficult to tell them apart, eventually.
Meanwhile, to answer your question, I'll quote StackShare:
FogBugz belongs to the "Issue Tracking" category of the tech stack, while Azure Boards can be primarily classified under "Agile Project Management".
Some of the features offered by FogBugz are:
- Issue and Bug Tracking
- Effortless Bug Reporting, Customizable Case Lists, and Full Case Histories
- Wikis, Evidence-based Scheduling, Task Outlining, and Reporting
- Email Management, AutoSort, and Snippets
On the other hand, Azure Boards(*) provides the following key features:
- Track all your ideas at every development stage and keep your team aligned with all code changes linked directly to work items.
- Scrum ready - Use built-in scrum boards and planning tools to help your teams run sprints, stand-ups, and planning meetings.
- Built for insights - Gain new insights into the health and status of your project with powerful analytics tools and dashboard widgets.
(*) Boards within Azure Devops mimic the functionality of Atlassian's Issue & Project Tracking software, Jira. Within Boards you can create kanban boards, story items, feature items , backlog items, sprints and query items to query your entire project items.