Technically both SIP and Skype based services are VoIP.  There are some important differences listed below.

Skype relies on your computer being on as it runs off a software program you install on your computer. 
You can use skype just like a VoIP softphone, simply with a microphone and headset attached to your computer or a USB phone. You always need to have the Skype software application running and your computer on.
There are now cordless phones available so you are no longer tied to your computer for cheap skype calls. You can wander away from the PC or laptop talking on Skype, but these devices (DUALphone, Philips 321 and Linksys CIT200) do need to be plugged into your computer via the USB port and have the software running. These phones also work with your existing PSTN number, so calls can be accepted from your normal numbers and from skype contacts.
VoIP in its simplest form runs on your computer with a softphone, similar to Skype.
Making calls over VoIP with a VoIP provider and an ATA or IP Phone means you can do away with the computer and roam your house/office easily, and integrate with your existing home line seamlessly. VoIP does not require your computer to be on, or for you to have a computer, just a fast internet connection.
VoIP supports open protocols agreed to by providers. With a VoIP ATA or IP Phone you can utilise the open protocol to change providers or use multiple providers to take advantage of cheaper rates for different call types (mobile/long distance/local).
SkypeOut allows you to call numbers other than Skype subscribers. This is usually is a charged call (Skype to Skype calls are free - same as most VoIP to VoIP calls for subscribers of the same VoIP provider).
For the security conscious, one of the advantages of Skype is that it is encrypted, while currently most VoIP calls are not.
Because you are using the software online you are able to view other Skype subscribers online.

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