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What are the differences?
How do you implement either in FPSC?
EfxMod - far better perspective mapping than I could dream of; however, I have difficulty seeing how it functions.

My work on darkbasic (using modified normals and heights to reduce artifacts; this is parallax)
Used slightly modified parallax shader from 2006 ultimate shader pack; for a truly impressive usage IN THE FPSC CONSOLE - yes, right in your FPSC game - check out the efx mod 1.5. I am eagerly awaiting 1.5 where they show it working without a hitch; the only reason for my work is to learn.
Update:
Wait, that's relief mapping in Efx. It's an evolution of parallax I think; I think they use only parallax in Oblivion, some mods out there convert it to steep parallax.
I suspect parallax is a PART of relief, included in the self-shadowing; they use that in HL2 EP:2

http://www.inf.ufrgs.br/~oliveira/RTM.html
Now this - this is absolutely impressive. It seems I'm going to have to use this as the standard for any further work.
It shocks me that this was around all of the way back in 2005 - and it takes so long to implement this impressive technology. Mayhap the relief mapping is actually changing the mesh geometry.
Thus far, this is the hierarchy:
Bump - Normal - Parallax - Relief
Bump merely changes light absorption patterns
Normal alters the angle of reflection, etc. for said light
Parallax gives the illusion of depth by further making use of normal and height maps
Steep parallax fixes some of the artifacts created by parallax but does not fix silhouettes
Relief actually alters the interpretation of the mesh with use of a UV map; like parallax, but actually changes the mesh without requiring a special mesh be tailor-made; many advantages - theoretically infinite resolution, no sharp edges, processor usage saved by fewer polygons
Disadvantages?
Mayhap it uses more ram and processor space through additional computations; I suspect most hardware could handle it; most excessive processor usage seems to be caused by shoddy game design (see: Oblivion, Simcity Societies - beautiful graphics, I get the feeling there could be more done to cut back on the RAM used by the texture and full-screen effects)
On that note, I love DarkBasic - it feels like Pascal with easy 3D; could basic be streamlined as it is translated into assembly language to add efficiency? It takes a while for textures to load in, I'd like to use higher-res textures for high-definition.
On which note, above screenies use 1024x1024, but it's hard to tell with the parallax mapping curving things quite nicely.
It seems relief would be absolutely required in any future game - it is absolutely essential. I will continue to focus resources there.

Download this if you haven't got Photoshop or Paint Shop Pro with the Nvidia plugin

http://www.crazybump.com/

The EFX relief shader does look very nice. I just hope it works 100% with segments unlike the stock bump mapping shaders.

What many people fail to realise (not sure if you have or not) is that many shaders (including bump/relief/parallax mapping) require a second texture to tell the shader how it should alter the appearance of the texture when viewed in the game.

In FPSC, this texture file is usually notable from the _N it has at the end of the file name.

Shaders do not automatically understand what parts of a texture you want to be relieved and what parts should be shown higher.


The one and only,
~PlystirE~

Yes, the system with which it is read from is flawed, however. If you have a normal texture, it would read it as '_N.TGA'; this is overcomplicated and forces you to choose one file type.
Efxmod succeeds in that it redoes the shader system... FPSC is wonderful, excellent software. It is absolutely brilliant, it only suffers from some minor yet fundamental, crippling flaws. It's basic, it's efficient - it just needs to be optimized - stripped down to provide the common user with more control.
I'm done rambling now, here is a video of my current progress - I use the exact same parallax shader provided by the 2006 free shaders pack, I just use better normal and height maps.
http://youtube.com/watch?v=glNopdsCSr0
I created the normal and height maps with crazybump; thanks to Nomad Soul for the link.
Summary:
Efx shader seems to redo the shader system. It is very, very versatile on that front; it also allows for beautiful relief mapping. Search youtube for a video of that and wait eagerly for v1.5... I sincerely hope v1.5 will address some minor flaws and perfect its integration with FPSC.

Now this is extremely convenient. Right on that site I have found a few .fx files. They CAN run in DarkBasic and are compatible with both Nvidia and Radeon drivers. Very nice, would appreciate further improving compatibility - I can make it work to some degree in DarkShader and it doesn't crash when I load it in DarkBasic - it's just all black. Here is the link:
http://www.inf.ufrgs.br/~oliveira/RTM_files/relief_mapping_Fresnel_fx.zip