What does changing Max Server Memory do besides clear the plan cache and (obviously) alter the memory setting)?
Ref: Server Memory Server Configuration Options

max server memory controls the SQL Server memory allocation, including (essentially any memory clerk found in sys.dm_os_memory_clerks):


the buffer pool
compile memory
all caches
qe memory grants
lock manager memory
clr memory


So all of these components will be effected by max server memory setting change.

On a side note this will be a very difficult path to find what is causing your CPU spike. I suggest you use following articles for your troubleshooting.


Troubleshooting SQL Server CPU Performance Issues by Joe Sack
How do I find out what is hammering my SQL Server?