There might be legal or privacy concerns.
First of all, a stored consciousness would functionally be identical to the living consciousness at the time of backup. Understandably, an awake copy might want to do some things to pass the time (watch space netflix, play games, gamble, etc) which would be a real problem for the living individual. If the backups were awake, there'd be lawsuits in no time concerning who is the rightful owner of certain items purchased in that time. There'd be people sueing their own backups, people freaking out because someone "hacked" into their bank account and so on. And of course there'd be the 'smart' ones who get a backup taken, change their important passwords, get killed and then figure out that the backup can't access their assets.
All of those are good practical reasons why you wouldn't want the backup awake.
But privacy is a better reason (or a different one at least). Imagine if your backup was awake and someone malicious (a branch of the government, criminals, a sleazy insurance salesperson, your mother-in-law, etc) got access to your backup and started pulling data from it. Doesn't even have to be anything dramatic. You go in for a backup, the machine turns on and next thing you (backup you) know, you're in a room basically being interrogated or even tortured for days upon days untill you tell the man in the black suit everything you've ever done wrong. Once they have a nice big list of things to fine (or worse: 'behaviorally correct') you for, they delete the interrogated backup, restore the previous one and BAM, you find yourself on the wrong end of the government's sights.
Same could happen with criminals extracting passwords, blackmail data, security features for your work, etc, etc.
For those reasons, I imagine that human rights activists, privacy advocates and other interest groups would strongly lobby for heavy encryption on backups and likely also try to push through rules and regulations to ensure backups remain 'asleep' untill they are needed.