IMHO both take a similar personality to preform the tasks that the jobs would consist of.
Database
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Troubleshooting and lots of patience
Extreme attention to detail
Will work with some hardware but probably mostly dealing with software
You will probably not be rewarded for things working only called out when they are not
Probably will not need to be as genius with numbers as a networker depending on the type of database you are working with
If the database goes down it should be the networkers job to unsure your backups are implimented so you can rest easy while the buck is passed to him in this situation
Network
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Troubleshooting and lots of patience
Extreme attention to detail
Will work with both hardware and software on a regular basis
You will probably not be rewarded for things working only called out when they are not
Need to have excellent math skills
Most likely resposible for any problems with the network including connectivity, outages, security. If you specialize in security you could be looking at a healthy paycheck.