Outbound mail fails after upgrading to 4.7.1
From what I understand, this can be because the server you contact has a self-signed certificate (see this question for a workaround), or because the PEAR SMTP package version used by CiviCRM does not support TLS. Civi uses 1.6.1, but support was added in 1.7.1.

In our case (php7, CiviCRM 4.7), we are able to work around this by forcing SSL connection rather than TLS. In the Outbound mail settings
1) put ssl:// in front of the mail server name
2) change the port to something appropriate. Usually to 465**