Account statements and the account information provided by your personal finance software should be coming from the same source, namely your bank's internal accounting records. So in theory one is just as good as the other. That being said, an account statement is a snapshot of your account on the date the statement was created, while synchronizations with your personal finance application is dynamically generated upon request (usually once a day or upon login).
So what are the implications of this?
Your account statement will not show transactions that may have taken place during that period but weren't posted until after the period ended (common with credit card transactions and checks). Instead they'd appear on the next statement. Because electronic account synchronizations are more frequent and not limited to a specific time period those transactions will show up shortly after they are posted.
So it is far easier to keep track of your accounts electronically. Every personal finance software I've ever used supports manual entries so what I like to do is on a daily basis I manually enter any transaction which wasn't posted automatically. This usually only takes a few minutes each evening. Then when the transaction eventually shows up it's usually reconciled with my manually entered one automatically.
Aside from finding (infrequent) bank errors this has the benefit of keeping me aware of how much I'm spending and how much I have left. I've also caught a number of cashier errors this way (noticing I was double-charged for an item while entering the receipt total) and its the best defense against fraud and identity theft I can think of. If you're looking at your accounts on a daily basis you're far more likely to notice an unusual transaction than any monitoring service.