Most of my experience is with Business Objects. I have done a little with Cognos Elite also. Comparatively I find Business Objects much easier to work with when it comes to customizing the presentation of the data. Neither of the two products does a drill-down as well as you can in Crystal Reports, but they both allow you to create multiple views of the data on one page. Business Object WEBI reports allow you to use multiple universes and multiple report pages in one WEBI document.
Both system will force you to use another layer between the presentation and data layers. In Cognos it is called Framework Manager. In WEBI it is called Universes. Both Universes and Framework Manager define the structure and connections the are usable to the presentation layer. Both middle layers can connect to any kind of database or even spreadsheets on an accessible drive.
Both products will force you to use their WEB portals as an interface into the presentation layer. Both allow you to schedule and deliver reports in multiple formats and in a variety of means. Links to reports, send the reports via e-mail, only send reports with data the user can see, and send alerts when reports fail are just a few of the common things that can be done.
When it comes to ease of use, Business Objects is a lot more intuitive and straight forward. Cognos seems to make you take a lot more steps to get the same results. Some of those steps are tough to figure out. The term "clunky" comes to mind when I think of working in Cognos.
Both products offer very similar concepts and tools to summarize and chart the data in a report.
Licensing costs and how you want to architect your server infrastructure are going to be the key driver to which product fits your organization better. Business Objects offers the advantage of another more static reporting tool with Crystal Reports.