Sending and receiving government business e-mails using private e-mail servers and clients is not just borderline criminal, it is a criminal act, a violation of Federal law. (Although the President may be exempt from those laws; have to check that . . . ) Government E-mail communications is "point-to-point" (even if it has a lengthy CC list attached) and often carries classified information not to be disbursed to "the Public". For those reasons it needs to stay within the government's "closed" (and highly secured, we hope) network of e-mail servers. That's no different that many large companies. (I worked for a company that encrypted every e-mail through their e-mail clients and servers and then had a second level of security for very "sensitive" information that even the NSA couldn't "crack" quickly.)
Twitter on the other hand is known to be a "Public Forum" with a multi-point distribution (everyone that cares to look). There is no expectation of privacy, quite the opposite in fact. Items are posted to Twitter with the expectation that many, many people will see and read what's posted. In the case of government officials it's like a press conference without the press "filtering" what's said (until after the fact).