What are the pros and cons? Why should we use 2013 instead of 2010? Are there any significant changes that you will see, or is it just a design fresh-up and some minor change!
It's quite a wide question to answer, but you could always start to read the article Changes from SharePoint 2010 to SharePoint 2013.
Personally I think there are extensive improvements in Enterprise Search (Content processing and Web Anlytics) as well as the built in Social capabilities of SharePoint 2013. The new social intranet and the improved search would make me decide on 2013 instead of 2010.
Here are some other examples...
Enhancements for Developers
Enhancements for Admins/IT PRos
SharePoint 2013 Feature Comparison Chart - All Editions: A table formatted summary of the all the features available in each edition of SharePoint 2013 (Foundation, Standard, and Enterprise)
Discontinued Features: Discontinued features and modified functionality in Microsoft SharePoint 2013
Machine Translation Services: create and configure Machine Translation services in SharePoint Server 2013
As Benny Skogberg said Search and Social got big improvments - I totally sign that.
Additionally I would mention the possibilities you have with the new design concept (device channels, mobile support, design manager, display templates and all the new stuff).
Another big thing is that the internet license is gone - its much cheaper now for small companies to use *one platform for intranet and interne*t.
If you have the choice, on a functional focus I would definitly emphasize on SharePoint 2013. For example : SPF 2013 now has some "server" functionalities for search.
On a technological focus, SharePoint 2013 is much more hungry.
But if SP2010 native functionalities are enough four your business needs, it may be a good choice.