The invokeapplicaton method can invoke only an executable 
application(file with EXE extenstion)

The systemutil.run method can invoke any application.

For example

systemutil.run "c:\testdata.txt"
invokeapplication "c:\diskcleaner.exe"

Invoke is used to exeecute an .exe file but by using 
systemutil.run we can open both .exe file as well as 
browser fles.

invokeapplication "c:\IE\Iexplore.exe"

SystemUtil.Run "c:\IE\Iexplore.exe", "www.yahoo.com"

I think i am right. Please correct If I am wrong.

SystemUtil.Run uses for web based application 
for ex: "WWW.ebay.com"
invokeapplication uses for windows based application 
for ex: "c:Windows\folder\subfolder\application.exe"

Invoke application is used to invoke any .exe file, where as
SystemUtil.Run works with any system utilities like
“Iexplore” apart from .exe file

for more details logon to: 
http://ssofttesting.blogspot.com