This was likely done to avoid requirements to turn over emails in cases of FOIA requests, or I think a few other conditions. There's supposed to be some degree of transparency in government communications.
By itself, that alone isn't too bad, I think. But the other reason why government officials should use government infrastructure for communications is security.
Messages that were classified after-the-fact (not at the time) were found to have been sent to/from Clinton's private server. I believe some messages that were classified at the time were also there, but were improperly marked, so someone could plausibly claim they did not know they were classified.
EDIT: There were in fact many emails properly marked as classified at the time. Some marked top secret. This definitely demonstrates carelessness. http://thehill.com/policy/national-security/286513-fbi-direc...
Further, after she received a subpoena from the FBI to turn over all emails on her server, 33,000 emails were allegedly deleted. This doesn't necessarily mean anything, but could indicate a cover-up of something shady or even illegal.
For the most part it's just a lot of speculation and character judgment. Even if this second investigation doesn't find anything illegal, it shows someone trying to shield themselves from public scrutiny at best, or potentially trying to cover things up at worst.