When comparing TortoiseGit vs GitHub Desktop, the Slant community recommends TortoiseGit for most people. In the question“What are the best Git clients for Windows?” TortoiseGit is ranked 9th while GitHub Desktop is ranked 17th. The most important reason people chose TortoiseGit is:
Licensed under GPL.
Specs
Ranked in these QuestionsQuestion Ranking
Pros
Pro Open source
Licensed under GPL.
Pro Windows context menu integration
Context menu enables access to common commands quickly.
Pro Can parse and provide a pretty log of all commits or filter by branch
Pro Convinient blaming tool
Very friendly blame tool. Easy to walk in the history of a file.
Pro Great GitHub integration
This is the official GitHub desktop client built by the GitHub team.
Pro Simple, streamlined GUI
GitHub Desktop uses an extremely simplistic two-panel view. It's not capable of complex historical visualisations like other GUIs, but it is very easy to use (especially for git novices).
Pro Supports pull requests
In addition to being able to seamlessly and easily integrate with all of GitHub's features, it also supports forking and submitting pull requests on any open source project hosted on GitHub.
Cons
Con Windows only
No Linux or OSX versions available.
Con Renames git commands
Makes things hard to find for people used to the git CLI.
Con Buggy file status icons
The file icons are also often buggy and do not reflect its true status. Often times the icon is missing and makes you think a file or folder is untracked, when it is already staged, or sometimes even already committed and pushed.
Con May clutter your Windows Explorer
If you have synced your dev folder to a cloud service, TortoiseGit's git status file icons will override your cloud provider's icons.
Con No support for staging
It does not support staging in any way. You'll never guess that this feature is exist in git.
Con Limited
Can't handle complex tasks. The Help Manual advises to use command-line Git instead.
Con Does not support multiple Remotes for a repo
Only allowed to assign one URL as remote. To manage/sync/fetch other remotes, use command-line Git instead.
Con Overly Simplified UI
UI that is designed not to support the needs of power and enterprise users. Management of more than five repos is next to impossible.
Con Buggy
Poster child for authors' programming ideology (FRP), likely the cause for the odd quirks and bugs it has.

Con Not free/libre
This application is proprietary, and thus cannot be modified or freely distributed.
Con No Linux support
There's no Linux version of this client.
Con Non-GitHub repositories are not fully supported
Since this is mainly a GitHub client, other repositories are not fully supported and with as many features and setting up a repo hosted anywhere else but GitHub is troublesome.
