I haven't worked as much with QGIS as ArcGIS. ArcGIS has been around for a very long time and it is versatile in terms of its computational capabilities especially in the geostatistics arena
QGIS is a desktop GIS software like ArcGIS, major difference between QGIS & ArcGIS is that it is opensource and a well match for ArcGIS, it performs most if not all works done by Arc, I am not familiar with Esri Web platform but I know there is a cost to it hence my decision to use the FOSS. I use Arc GIS and QGIS
I am a social scientists, not a developer, so I use GIS to manage spatial information, analyze it, and visualize findings. I use QGIS in the classroom as well as ArcGIS online to visualize spatial data. If I have a research grant, I often buy a network license for ArcGIS to manage spatial information. For spatial statistics, I usually use the spatial packages in R (e.g., sp, rgdal) or pysal. I suppose you could clarify what you mean by "Web GIS"?