You probably should choose just one of them to keep.
Anti-virus programs are the same way, most of the time. If there are two, they'll always be interfering with each other. Things like detecting viruses in the other one's quarantine file, or detecting the other program as a virus.
Just keep one.
And here's why:
In order to work effectively, antivirus/spyware programs have to jump in and scan files before Windows (or whichever operating system) opens them. If you have one scanner then this is fine - the program jumps in, looks up its virus definitions, scans the file, determines it's safe, then finishes and passes control back to Windows.
If you have two scanners then this becomes effectively an arms race between the two programs. Say Scanner A gets in first. When it goes to run or read its virus definitions it also has to open files. Normally this wouldn't be a problem because it would know it is safe. However, Scanner B does not so it tries to jump in and scan Scanner A's files. Of course, in order to do this Scanner B needs to run and look at its own defintions. Which triggers Scanner A....
This can continue until one program gets lucky enough to finish and basically degenerates into a hideous mess where each program keeps jumping on the other. Obviously this is rather bad for performance!!!
They are right.
I personally have used both in the past. And I recommend Spybot to friends, family and customers.