First things first, you are slightly misusing the term Virus and since you said you wanted to learn I mine as well teach a little extra.
If you imagine types of malicious software as a tree, the very top of the tree (root node) is malware. Basically its the words malicious software gelled into one word. A virus is a form of malware that is defined by the way it re-produces. Viruses are defined as a virus because they can transfer themselves from host to victim in almost anyway including over a USB stick, email, internet, program download etc. and is a very generic term.
I read the wikipedia page for crimeware, and to me it seems that crimeware is just a specific usage of spyware.
You are correct, what defines crimeware is exclusively the intention the attacker has for creating their malware. For example, a Trojan horse could be crimeware. The Trojan is the delivery mechanism (Hey look at this super cool flashlight app! Wow, so cool. Why does it need Internet permissions and file permissions? Ah don't worry about it.) but the real reason for creating this flashlight app you can only download off a third party website is because it is a Trojan horse delivery mechanism for some crimeware key logger. The flashlight app may contain a key logger that transmits back to a command server it's findings including all the passwords you type in.
This of course would be different than a Trojan horse flashlight app who's end goal it is to encrypt your entire phone's drive and demand ransom (Ransomware).
Crimeware is just a spin-off of Spyware and I imagine (although I have not first hand analyzed examples for comparison) that Spyware and Crimeware are near identical in how they are written, but may be delivered and behave slightly differently as they have different goals. For example, Spyware includes things such as watching people on their webcams whether that be for corporate espionage or for perverted reasons, you choose, both exist.
Is there anything a network administrator needs to know specifically to defend against Crimeware that isn't included in plain-old spyware?
I would say no, not really. Crimeware is just a more tailored, targeted type of Spyware and the same sort of precautions should protect you. Keep un-needed ports closed, do not download un-trusted files, be careful with phishing emails, avoid putting potentially infected USB's in company computers and keep personal devices off the network.