Total connect is the smart phone/internet app to remote controll the unit. Arm/disarm, see sensors statuses, get alerts, emails, etc. This will require the unit be connected to a monitoring service via GSM dialer, wifi module, rj45 network module. This also can only be enabled by a monitoring company. There is no way for you to self monitor and access total connect for free.
It also will log all activity so you can go back and see when sensors were tripped even when the alarm isn't on. In one of the recent upgrades to the iphone app to they enabled some geofencing functions so if you leave the house for example but tc see's the alarm isn't active and the phone is out of the neighborhood it will alert you.
I don't have total connect at home (it's normally around $10 more a month ontop of your standard monitoring) but I have it on one of our business buildings so I get an email when the last person locks up and arms it for the night. If I don't see that email by 11 or midnight is when I start hitting cameras up to see what's going on. For home, the wife and I are both so good about arming it it's not really needed, and it's so easy to set user codes I have two extra codes programmed so if someone ever needs in the home I can give them one of those and just delete it once I get home.
See some screen shots below off my iPhone