firstly, it's not just about performance, there are other reasons for upgrading.
Security - There are no known security issues in Joomla 1.5.26 however 2.5 and 3.2 (especially 3.2) have far better security.
Support - Support for Joomla 1.5 was stopped in September 2012 and I believe Joomla 2.5 will continue to be supported until December 2014 and Joomla 3.x untill December 2016.
Features - Joomla 3.2 contains so many more feature such as
- Mobile Responsive
- Ajax Interface integrated into the core
- Ability to install extensions via the backend
- and much more
Performance - To simply answet this question, yes, Joomla 3.x performance is better than Joomla 1.5. I couldn't find a raw demo for Joomla 1.5 but here is a speed test of a raw Joomla 3.2 site:
Do also bare in mind that there are lots of extensions out there to boost the performance even more by compressing and gzipping scripts/stylesheet and so on.
And out of all honesty, I do believe that if something works then why change it (in some cases) but you also have to understand that things are moving forward, such as new web technologies are being introduced which will help improve your site.
Overall, I would most definitely upgrade your site to Joomla 3.2
Hope this helps