Similar

Both are free and open source web applications with a powerful wiki.
Both can run on standard, inexpensive hosting.
Both support Wiki markup. Please see (if) Why Wiki Syntax Is Important to your project.
Both are written in PHP

Different

DokuWiki uses files for storage whereas TikiWiki uses a database, such as MySQL. Because of this, DokuWiki is easier/quicker to install.
While people often compare them, TikiWiki is more comparable to Drupal and Joomla! as it's a full-featured CMS which happens to be Wiki-centric.
DokuWiki is focused on being a great Wiki (which it is!), whereas Tiki is more of an out-of-the box Content Management System / general purpose wiki with all the features built-in. Because of these design choices, TikiWiki has more features out-of-the-box. DokuWiki doesn't offer forums, bug tracker, blogs, etc
DokuWiki is GPL, TikiWiki 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.