Node Method vs Simple KVL and KCL Equations Difference
Firstly, I believe you have a sign error in your second KCL equation (although I'm having a little trouble reading it, so I could be wrong). 

Assuming your algebra is correct, surely the node voltage method is equivalent to KCL/KVL. Look at your node voltage equations: they're simply using Ohm's law to determine all the currents passing through each node, and then the equations you get correspond to identically those from just applying KCL (although they're written in terms of voltages and resistances as opposed to currents). 

From experience, I'd say with the circuit you've shown here, Mesh Analysis would probably be a little more efficient than Node Voltage Analysis, since your circuit only has one super-node and a bunch of other nodes that can't be clustered. 

Having said all that, Node Voltage Analysis and Mesh Analysis are simply systematic methods of applying KCL and KVL. They're basically algorithms that prescribe steps for analyzing arbitrary circuits.