1. Syntax and Language: Elm is a functional programming language that compiles to JavaScript, focusing on being simple and providing strong static typing. Pug, on the other hand, is a template engine that simplifies HTML coding by using indentation to define structure. Elm requires learning a new syntax and functional programming concepts, while Pug mainly focuses on simplifying HTML structure with its concise syntax.

2. Compilation process: Elm source code needs to be compiled into JavaScript before it can be run in a browser. This compilation ensures static type-checking and eliminates runtime errors. In contrast, Pug templates are rendered on the server to generate HTML for the client, with no explicit compilation step needed like Elm.

3. Data Management: Elm comes with its own built-in architecture, the Elm Architecture, for managing application state and data flow effectively. This architecture enforces a unidirectional data flow, making it easier to manage complex frontend applications. Pug, being a template engine, is primarily focused on presenting data passed from the server and does not offer built-in data flow management like Elm.

4. Tooling and Ecosystem: Elm has a focused ecosystem with tools like Elm Test and Elm Format for testing and code formatting. It also has a strong package and dependency management system through Elm Package. In contrast, Pug has a simpler ecosystem with tools like Pug CLI for rendering templates and does not have an extensive package management system like Elm.

5. Error Handling: Elm is known for its helpful compiler error messages that guide developers in fixing issues at compile time, reducing runtime errors significantly. These error messages are precise and informative, making debugging easier. Pug also provides error messages but is more focused on template rendering errors than static type errors like Elm.

6. Performance: Elm is designed to create highly performant web applications due to its static typing and functional programming principles. The compilation process optimizes the generated JavaScript code for efficiency. Pug, being a template engine, focuses on generating HTML efficiently but does not have the same level of optimization for runtime performance as Elm.

In Summary, Elm and Pug differ in syntax and language usage, compilation process, data management approach, tooling ecosystem, error handling, and performance optimizations for web application development.