There are numerous markdown engines and implementations, of which GitHub Flavored Markdown and MarkdownPad are just two. While they all implement markdown with basic similarities, there are always differences.
From looking at your screenshots, it seems the main issues lie with headings and code formatting.
For headings, try adding a space after the hash symbol:
For code blocks, try wrapping the code with three backticks:
``` Foo ```
In short, your best option is to read the GitHub markdown documentation and adjust your markdown text to fit with GitHub's formatting and syntax style.