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
The minimal download strategy will use a single .aspx file (start.aspx) for your pages, with the actual URL encoded in the text following the hashmark (‘#’). When moving from page to page, only the changes between two compatible pages will be downloaded. Fewer bytes will be downloaded and the pages will appear more quickly.
BCS: OData connector (Standard/Enterprise)
The OData connector is new for SharePoint. It allows for Business Connectivity Services (BCS) to use a RESTful OData endpoint as a data source for external lists, Business Data WebParts, and custom user interfaces.
BCS: Alerts for External Lists (Standard/Enterprise)
SharePoint now provides the capability of using alerts for external lists, just as they have been used for traditional lists. A user can subscribe to be alerted when data changes on an external list. Learn more about external events and alerts in SharePoint 2013.
Enhancements for Admins/IT PRos
Not available to SharePoint Online customers. Shredded storage is essentially the chunking of data—only differences are saved. That means if versioning is enabled and someone makes a change to a document, only changes “shreds” are added to the storage footprint of that document. Shredded storage improves I/O performance by reducing how much information retrieved by the web server from the content database. Shredded storage removes duplicate files, and improves data transmission speed.
Not available to SharePoint Online customers. SharePoint Server 2013 customers can use the Distributed Cache service to cache feature functionality, which improves authentication, newsfeed, OneNote client access, security trimming, and page load performance. Learn more about Distributed Cache.
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
One huge thing, IMHO, is the licensing change. External users no longer cost you CALS, and they removed the extra fee for running a public internet site in SharePoint 2013. Saves TONS of money vs the old setup.
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.
SharePoint 2013 has a complete refurbished look than its predecessors. It has a metro style touch friendly user interface. There are some really good features added such as the drag and drop feature. Plus, you can extend its functionality using SharePoint Apps, which is quite easy to use even for laymen. The social features are also a great improvement in SharePoint 2013. You can check the social features changes in SharePoint 2013 in the following video. https://www.youtube.com/watch?v=JPa_v3N4oGE&list=UUt7QGtx3CkA0sKQml-reiuA.