There are heaps of differences between cloud-based apps and traditional ones, all of which apply to accounting suites as well. While it depends on the specific solutions you're comparing, key advantages to cloud-based apps include mobility (access your data anytime, anywhere), scalability (the software grows as you do and doesn't require on-premise maintenance), and usability (next gen apps are generally way more intuitive than their on-premise peers).
In terms of security, it totally depends on the apps you're comparing. Most of the best accounting apps (Xero, Quickbooks online, Kashflow, Expensify, etc...) use banking-grade data encryption.