It appears the answer to the thread title is "not much".
I could go on for pages listing the types of things the spu's are used for to make up for the machines aging gpu, which may be 7 series NVidia but that's basically a tweaked 6 series NVidia for the most part. But I'll just type a few off the top of my head:
1) Two ppu/vmx units
There are three ppu/vmx units on the 360, and just one on the PS3. So any load on the 360's remaining two ppu/vmx units must be moved to spu.
2) Vertex culling
You can look back a few years at my first post talking about this, but it's common knowledge now that you need to move as much vertex load as possible to spu otherwise it won't keep pace with the 360.
3) Vertex texture sampling
You can texture sample in vertex shaders on 360 just fine, but it's unusably slow on PS3. Most multi platform games simply won't use this feature on 360 to make keeping parity easier, but if a dev does make use of it then you will have no choice but to move all such functionality to spu.
4) Shader patching
Changing variables in shader programs is cake on the 360. Not so on the PS3 because they are embedded into the shader programs. So you have to use spu's to patch your shader programs.
5) Branching
You never want a lot of branching in general, but when you do really need it the 360 handles it fine, PS3 does not. If you are stuck needing branching in shaders then you will want to move all such functionality to spu.
6) Shader inputs
You can pass plenty of inputs to shaders on 360, but do it on PS3 and your game will grind to a halt. You will want to move all such functionality to spu to minimize the amount of inputs needed on the shader programs.
7) Msaa alternatives
Msaa runs full speed on 360 gpu needing just cpu tiling calculations. Msaa on PS3 gpu is very slow. You will want to move msaa to spu as soon as you can.
8) Post processing
360 is unified architecture meaning post process steps can often be slotted into gpu idle time. This is not as easily doable on PS3, so you will want to move as much post process to spu as possible.
9) Load balancing
360 gpu load balances itself just fine since it's unified. If the load on a given frame shifts to heavy vertex or heavy pixel load then you don't care. Not so on PS3 where such load shifts will cause frame drops. You will want to shift as much load as possible to spu to minimize your peak load on the gpu.
10) Half floats
You can use full floats just fine on the 360 gpu. On the PS3 gpu they cause performance slowdowns. If you really need/have to use shaders with many full floats then you will want to move such functionality over to the spu's.
11) Shader array indexing
You can index into arrays in shaders on the 360 gpu no problem. You can't do that on PS3. If you absolutely need this functionality then you will have to either rework your shaders or move it all to spu.
Etc, etc, etc...