I’ve been working with the three applications that have been mentioned so far (G.Projector, Flex Projector, Geocart) quite a bit, so here are a few comments:
Both G.Projector and Flex Projector are great free tools.
However, both of them do not have the greatest interpolation, so I recommend to export the image larger than actually needed – e.g. if you need an image of 2000 x 1000 pixels, export them using the size 4000 x 2000. Then, use an image processor to scale them down using a decent interpolation method. I’m usually using Lanczos-8 or Catmull–Rom on the Natural Earth raster images.
Using Geocart, this isn’t needed: When you select appropriate quality settings, you can use the exported image right away.
Also note that G.Projector’s maximum output size is limited to 7500 x 3750 which might not be enough if you want a really large map for high quality print.
Flex Projector offers a comparatively small set of projections – which, of course, isn’t very important as long as the projections you want/need are included.
G.Projector and Geocart both offer more projections than you’ll ever need.
While G.Projector and Flex Projector can only provide the projected raster image, Geocart can do a lot more, e.g. you can use the Natural Earth shape files or add labels for countries, cities etc. But keep in mind that on most occasions, you’ll have to post process such maps, e.g. you’ll have to re-position the labels or out them on a vector path. Fortunately, the PDF files exported from Geocart, can be opened using applications like Adobe Illustrator (so the manual says), Affinity Designer or PhotoLine (the latter two I tried myself).
So for a global map, you probably don’t need Geocart + a GIS application, but Geocart + a graphics application.
Granted, Geocart is quite expensive. I think it’s worth the price, but of course you’ll have to decide that for yourself.
If you just want the Natural Earth raster images to be transformed to a different projection, both G.Projector and Flex Projector work fine.
Flex Projector adds a .tfw file on export, neither G.Projector nor Geocart include GeoTiff data (and, as I’ve said above, you probably won’t even need them when you use Geocart).
Hope this helps,