What do you like best?
Inkscape is a solid tool and a standard when it comes to vectors. It's fully featured and provides all the needed tools for professional graphic illustrations. Exported SVG's come out clean and should work straight away on the web.
What do you dislike?
Unfortunately Inkscape has still no Print CMYK support, so if you want to prepare Print documents you need to import your files into another program which converts to CMYK, like Scribus.
Lack of Artboards makes working with multiple elements and pages quite tiresome, as sometimes you might want to export several assets at once, instead of copy-pasting them.
Currently the user interface is also a bit clumsy and not intuititive. Customizations are kind of limited and HiDPI support is stll lacking (especially on Linux.
Recommendations to others considering the product
Working with multi pages document is hard (PDF) as no artboards are supported. External software for merging pages is needed. User Interface is quite cluttered and non customizable. Some bugs in the line height settings when inserting type.
What business problems are you solving with the product? What benefits have you realized?
As a designer using Linux, Inkscape is one of the few, if not the only tool available for doing Illustration and UX Design. Vectors mean that it's lossless so you can prepare perfect documents with specific requirements. In the past, it has shown the Inkscape had the best support for SVG which work right away on the web. As open source software it is also independent from any company.