Illigegal Instruction means Steam (or some lib) tried to execute an CPU-Instruction your Processor does not know. There are two possablities for that:

1. The code has a bug and jumped into nowhere land where some data formed an illigal instruction.
2. Thanks to all the instruction extentions on the x86 architecture (SSE, SSE2, and so on) it could be your older AMD is missing something with is really compiled into the app.

Maybe as a first step you could attach the output of "cat /proc/cpuinfo" so developers could see what your CPU does support what it does not (Athlon XP is a wide range of processors, gone are the old days where a name cleary identified the model).

Here is what it says when I run that command in the terminal:

processor : 0
vendor_id : AuthenticAMD
cpu family : 6
model : 8
model name : AMD Athlon(tm) XP
stepping : 1
cpu MHz : 1250.000
cache size : 256 KB
fdiv_bug : no
hlt_bug : no
f00f_bug : no
coma_bug : no
fpu : yes
fpu_exception : yes
cpuid level : 1
wp : yes
flags : fpu vme de pse tsc msr pae mce cx8 apic mtrr pge mca cmov pat pse36 mmx fxsr sse syscall mmxext 3dnowext 3dnow up
bogomips : 2511.70
clflush size : 32
cache_alignment : 32
address sizes : 34 bits physical, 32 bits virtual
power management: ts

Ouch!
I think the problem is that your CPU does not have SSE2 support due to its venerable age.
This is not an error. It simply does not support these (then) new instructions.

http://en.wikipedia.org/wiki/SSE2

I guess the compiler used to compile the Steam client under Windows probably build an special detection and bypass scheme into the binary to prevent such problems.
Problem is GCC, the compiler normally used under Linux, can't do the same, so switching
off use of SSE2 means switching it off for everybody.