Files ending with 'w' (tfw, jpw or jgw, bpw and so on) are world files. These files are used to geolocate rasters that otherwise have no means of geolocation. They tell the GIS where the file's origin is in the real world and the size and rotation of the pixels. however, they do not store any information about projection.
You only need a world file if the georeferencing information is not already stored in the file header. Arc, QGIS and most other systems will store the information in the header and so, within the realms of GIS, the world file is not often needed as a separate entity. This is your use-case. Your files are georeferenced and projected and that information is stored in the file headers (based on your description) so you don't also need a world file.
So why have world file at all? World files predate the now common storing of georeferencing information in the image file headers. They have been around a long time but are less prevalent these days. The world file is a simple and human readable georeferencing method and is very useful where you don't or can't have the information stored in the header, or where you are doing something with the image, or are going to use it in a system that does not read and/or honour the geo-tags in the file headers (e.g. Photoshop, GIMP, some proprietary Wind Farm software packages and potentially many other systems too). Some image formats still can't store georefencing information in their headers (e.g. bitmap and also gif... if my memory serves me). To use such formats you still have to use a world file. For these files, the format is exactly the same, it is just the extension that is different.
The aux file is different. Although the aux file can and does store projection and coordinate information, it also stores information that is NOT normally encoded in the header such as raster statistics, colour maps and pointers to the pyramids file (.rrd). The aux file is a helper for Arc and is mostly useful for the raster statistcs these days (given that the coordinates and projections are mostly stored in the header). These precomputed statistics are used in many ways to speed up rendering and calculations. They are what give you the min and max values you see when you apply a colour scheme to your raster (for instance). QGIS also has an aux file but I don't know is a QGIS aux is readable by Arc and vice versa (i.e. has QGIS adopted the ESRI format as a common standard?).