Well, Bernoulli's equation is a very simplified form of the actual energy equation derived by using control volumes around the fluid flow considering all possible variations including time and space.
1) Bernoulli's equation doesn't account for any other form of work or energy other than potential energy, kinetic energy and pressure energy. For example, the fluid can flow through a pump doing work on it or a furnace heating a column of air. In these cases, you can't use the simple Bernoulli equation, as points before the area where work is being done on the fluid and points after it have different Bernoulli's constants.
2) Moreover, Bernoulli's equation between two points is valid along a streamline only. This is because different streamlines have different Bernoulli's constant (although, this is rare and is not true when the flow is irrotational)
3) For energy equation to be valid, the flow field need not be incompressible or non-viscous as is needed for Bernoulli's equation (unless you are using compressible Bernoulli flow equation and also adding the friction head to take care of viscous effects at the wall)
4) Further, the Bernoulli equation is generally used for steady and laminar flow. Energy equation written in its full glory takes care of variations in viscosity, density, pressure and velocity variation with respect to time and space. Written along with Navier Stokes equation, even turbulence can be accounted for.
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