Similar

Both are full-featured free and open source web applications with which you can realize a very diverse variety of projects.
Both are written in PHP and use a database such as MySQL
Both can run on standard, inexpensive hosting.
Both have large & vibrant communities
Both have a template layer

Different

Tiki is more centralized. Tiki is the all-in-one model - all modules are part of the package while Typo3 (like Joomla!) is the small-core-and-add-what-you-need model (you need to use third party modules out of the package). Each model has its pros & cons.
Tiki is more focused on collaboration and structured wiki (customisable data management) over the web while Typo3 is more focused on web publishing.
- The Typo3 community uses GitHub instead of dogfooding (working on your own tool), as confirmed by the banner at the bottom of their wiki subsite.
  - Please see (if) Why Wiki Syntax Is Important to your project.
Tiki is relying on the Wisdom of Crowds to shape its development model (while some people see it as the infinite monkey theorem instead). Consequently Tiki can suffer from a diffusion of responsibility since everyone only works on what interests them and not necessarily ensuring up to date documentation and basic reliability in some cases. Typo3 follows a professional open-source business model whereby people who contribute to the project can get compensated for their work.
Typo3 is GPL, Tiki is LGPL. The main difference between the GPL and the LGPL is that the latter can be linked to (in the case of a library, 'used by') a non-(L)GPLed program, which may be free software or proprietary software.

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