They both achieve a reproduction of what a real tube amp is doing, but they go about it differently.
A profiler (or capture), uses a microphone (one or more) and a test signal into the amp (Provide a known input, then measure the output of the speaker) to listen to how an amp behaves under different frequencies and at different volume level inputs. The profiler then knows how the amp behaves and duplicates this behavior when you play that patch with your own guitar.
A modeler attempts to match the behavior of the circuitry inside an amp with software simulations of what those circuits would do to the input signal.
They seem to be a world apart in approaches, but in fact all profilers/capture devices have models to modify the capture (the above post is incorrect that a profiler can only reproduce exactly the settings of the amp it was captured at). Some do it better or worse than others. The Kemper has been around quite some time, so it has gotten pretty good at capturing an amp, and providing controls to modify it after the fact as well. Older versions of the firmware did not allow as good of refinement of a profile/capture. This meant that you had to find someone that made a good capture of your VOX AC30 at or near the gain and eq that you wanted or it didn't work as well. This has been addressed and it works pretty slick now.
Modelers have the issue that they don't even attempt to "capture" an amp. They try to simulate what its circuit is doing. This leads to lots of "circuit" tweaking in the amp models to get a tone you want. They do "capture" the cab behavior though, so as you see, both approaches have elements of the other incorporated to get good results.
Kemper, Helix and Fractal all have very mature on-line libraries of canned sounds from different amps. Quad Cortex is much more limited as it is much newer.
At the end of the day, being an old tube amp guy myself (Fender HRD, VHT Ultralead, VOX AC30, Marshall JCM800), It is MUCH easier for me to setup the Kemper to achieve tones for cover songs than it is on a Fractal or Line 6 device. Furthermore, the foot controller for Kemper is very well thought out for live use as is the "Performance" mode that is expressly designed for gigs.
The Line 6 and Fractal approach both offer much more routing capabilities than the Kemper with regards to doing anything you want with as many effects as you can get your head around. They even offer using more than one amp model within a single rig (which Kemper can't do at all).
I rarely mess with my live setup. I have tones I use in general, and a few I use for specific songs (like U2 songs). Last night is the first time this year I have setup a new sound (Acoustic simulation for the intro for "Friends in Low Places"). Now that I have that acoustic sound, I will likely use it for any future acoustic song unless the tone is way off for that purpose.
It is my experience that people that love modelers LOVE to tweak. I love to gig
. For me, having a replacement for my tube amp rig is all I really want .... just lighter and less complicated to setup. I never wanted to be able to create a sound that my old rig could not, I only wanted to have a gig rig that was light (11.5lbs), fast to setup, and could cover the sounds I got with my old tube amps and pedal board.The previous poster is correct about browsing through the Kemper Rig manager though. You can spend days auditioning the different rigs others have produced and shared on their Kempers. That's pretty fun too, but I only do it when I am looking for a new tone I need to reproduce. I can generally find what I need in under 15 min and have it tweaked to my liking in under 30 min. It get saved in a performance (set of 5 rigs) that I will use for a certain set of songs and ..... done!
