A computer virus, by definition, is a piece of code which can replicate itself. Most viruses also carry a malicious payload or a bad joke. Spyware, on the other hand, is a kind of malware that collects information about users without their knowledge. In the past, vendors of antivirus programs often took their job literally and produced software which was able to deal with viruses only, leaving the other threats including spyware aside. These days, luckily, antivirus software vendors often ship their programs with built in antispyware functionality, yet it might be the case that you have pay for it as free AV software only protects against malicious programs which can replicate themselves, that is viruses. So, yes, you definitively need both, an antivirus program and antispyware protection, and probably even more!
When you purchase or obtain your antivirus software with ant-ispyware functionality are you safe then? Probably not. It is strongly recommend that you check whether your anti-malware program, which, as outlined above should protect against viruses and spyware, also protects your computer against root kits, keyloggers, Trojans etc. Believe me; some vendors play the semantic game with their customers so you had better look for a 360° anti-malware program which also protects against potentially unwanted programs (PUPs) discussed in Bright Hub’s article What is Grayware? As a best practice it is recommended not use to use mix products from different vendors of anti-malware software as they could be incompatible.
Exceptions to that rule are Windows Defender, Microsoft’s free anti-malware tool (one of the Top Ten Security Reasons to use Windows Vista and Windows 7) and Spybot Search & Destroy. Please note these two pieces of software can only complement, but not replace, your primary anti-malware program. As shown in our article Do I Really Need to Buy an Antivirus Program? Are Paid Products Better than Free Products? if you run a Windows operating systems then you probably have no option that buying a top Windows anti-malware program to achieve a high level of protection against all sorts of threats; Webroot AntiVirus with AntiSpyware is recommended. Besides all the threats which have been mentioned in this article does Webroot have an integrated URL scanner which warns you of scam and phising sites and, unless overridden, prevents you from entering web sites which infect computers with malware.
If you found this article interesting then you may want to check out What is a Zombie Computer? and How to protect from Ransomware.