I'll elaborate on my answer in the other thread. This is also the exact topic of my AU course this year...
Both systems are topology optimization algorithms, but their approach is dramatically different.
The more traditional approach is referred to as SIMP (solid isotropic microstructure penalization) which uses an FEA engine to determine what voxels are required (and which voxels are not) in order to carry the load. Technically speaking, the SIMP method looks at the voxels as partial density between 0 and 1 (0 being empty space, 1 being fully dense). Since this is (nearly) impossible to realize in all practicality, the penalization occurs which forces the voxel to either go to a density of 0 or 1. This is where the "load path criticality" comes from - When you run a shape optimization inside of Fusion you tell the algorithm what percentage of mass you want to remove. The solver then shows you a result at that reduction with the LPC set to 0.5. You can then adjust the slider up and down to show where each of the voxels falls in that partial density scale for the given mass removal amount. In other words, if the LPC is set to ZERO then the voxels present have at least a ZERO density. As that slider moves up, the voxels start to remove because they begin to have a calculated density lower than the LPC threshold. It becomes up to the designer to identify what level of reduction you're comfortable with (with the understanding that the 0.5 LPC result should carry the loads imposed during the optimization). This solve method (as currently implemented) can only find a single solution for the given domain - whether it is a global minima or just a local is a total unknown.
The level set approach (Generative Design) uses a different math model to achieve similar types of results. The math here gets more than a little deep. Essentially, the solver uses iterations (time steps) to move the surface boundary of the part in question in a normal direction to the load path while satisfying the parameter for minimal strain. There is a bit of "philosophy" here as well, that given enough iterations, the solver would be able to find the global minima of a design domain. I say "philosophy" because it's really an academic exercise rather than a practical reality. It quickly comes down to the designer's intuition as to whether continuing on is really worth it.
The math models aren't all that's happening here, though.
In reality, both methods are topology optimization. What makes Generative Design so different is a couple of features that other products don't offer:
1) Generative Design works specifically with interface geometry. You include all your bosses and flanges and then the first step that happens behind the scenes is that the design space is shrinkwrapped - to give a starting shape of the general envelope of your part. From there on, the two systems are quite similar in terms of (end) results. You do have the ability to provide specific starting geometry (see #3).
2) Level set methods are highly scalable across multiple processors. This means that tools like GPUs can crunch the numbers faster than a CPU would. This further manifests itself as being a differentiator when you begin to throw in multiple variables (e.g. multiple materials, multiple fabrication types). When you see Generative Design pump out tons of different results, it is because the solvers are actively chewing on different variables. In theory, you could do the same thing with a SIMP tool - but you'd be running individual instances on individual CPUs.
3) The level set method can be "hacked" to improve performance and to "guide" the solver to a specific solution. One of the math quirks of the level set method is that the more surface area a shape has, the faster the algorithm can remove mass. When we couple this knowledge with the ability to include specific starting shapes, we can then intentionally seed high-surface area models into the solver to improve the mass burn down rate. A second math quirk is that the solver can easily create (or merge) holes, but it struggles with where to put them. We can leverage this by including starting shapes with a large number of topological holes to get entirely different results than you might expect. Lastly, we can include exotic seeds (specific shapes) to guide the solver into a quasi-aesthetic. You may reach an acceptable answer based on your specific input shape - but it may not be the global minima - and that may be your intention all along.
I like to show (3) as an example:
Here is a design space where I want to create a shelf bracket.

If I don't include any starting shape, I tend to get things that look something like this:

But maybe I want to follow a more specific look. So, I add this as a starting shape:

And here is a result with that as a seed:

You'll see my comments around the forums that this isn't the black box where design goes to die that most people think it is. No, I don't have explicit control over the outcome... but the purpose of these softwares (at this stage of the development of them) is primarily about ideation and not end-design creation.
I'm sure I've left off info (the level set method is difficult to explain without more time/graphics/math). Keep asking questions, though (as it also helps me prep for AU!).
K. Cornett
Generative Design Consultant / Trainer