Norton is absolutely useless. As is symmantec and most any anti-spyware that will come pre-installed on your computer.
That said, it's best to have 2 or 3 different antiviruses/antispywares. Go online and download some free ones. I have AVG and MalwareBytes, which are both pretty good. I also have STOPzilla, but I'm considering uninstalling it. It's more of a pain than it's worth. Just browse around and read reviews.
Btw, to actually answer your question; no, you dont need to have an antivirus AND antispyware. Most of them actually do both and more, they're just advertised as one or the other. It is recomended that you have more than one, though. Because the one you have wont always detect every piece of virus, spyware, etc.
You need an anti-virus program (Avast, AVG, Panda)
anti-spyware (Ad-Aware, Malwarebytes, Spybot, Advanced System Care, etc)
and a two-way firewall (Zone Alarm, Ashampoo)
AND all of the above have FREE versions (as well as paid versions).
Norton, McAfee, NOD32, and many others have suites that offer all-around protections - for a price.
Incidentally, DO NOT run two different anti-viruses as they will conflict. Multiple anti-spywares, on the other hand, should compliment one another, but run separately.
And only one firewall.
Not all antivirus programs are gonna provide you with all the protection that you need, but there are other free installs that can assist what you have on your computer right now and maybe catch some stuff that it missed. I have Norton but it is a resource hog, so I have two other free download that I use that can scan everything as well and they don't conflict at all with what u have already installed. I was surprised after using these free download how much actual crap Norton missed, so good luck on this
norton really sucks. i had it, and it failed to pick up stuff. try kaspersky, download the free trial on the net. after that i just bought the full version.
I have use some form of Norton for ovre 25 years,never had a problem nor a virus,at present I am using 360 for about five years
Norton is a scam, it's bullsh*t. I use Malwarebytes & Avast.