Hint and Summary:
There are two key words and phrases you should ponder and read about in your quest to understand the difference: (1) Counterfactual Definiteness and (2) Bijectivity / "Reversibility" (said of transformations imparted to a system).
Even though I don't believe a one sentence answer does justice to this topic, a couple of one sentence answers might read:
- The key difference is that discussion of the Heisenberg principle admits no notion of counterfactual definiteness, whereas the observer effect for classical systems does; OR
- A blunt and less subtle difference is that the observer effect can in principle be made as small as one likes by doing more elaborate and careful measurement procedures, whereas physicists believe that no amount of experimental effort can do better than the bounds implied by Heisenberg uncertainty.
No. They are absolutely not the same thing.
In discussing the observer effect, measurement imparts a bijective transformation on a classical system's full state. Therefore, one can infer what the full system state was before the measurement in the noiseless case, and it is in principle meaningful to discuss what a classical system's state was before the measurement. Therefore, if two measurements do not commute in their effect on the system state, one can meaningfully infer what the measurements would have yielded if they were done in the opposite order from that actually made. Indeed, this "might have been" information can be usefully deployed in the design of control systems for classical systems.
In contrast, quantum measurement imparts a nonbijective mapping to a quantum system. Many quantum states can produce any given measurement. One can therefore not in principle "reverse" the effect of the measurement transformation to discuss what the quantum state might have been before the measurement. Even a prepared quantum system, in a known state, can yield different measurement outcomes when the same measurement is imparted to a the state. So there is no way one can talk about what outcomes two measurements would have yielded if imparted in a different order from what was actually done. The Heisenberg uncertainty principle simply quantifies this situation by computing the amount that outcomes from pairs of measurements imparted in different order differ from one another as a statistical variance computed from the commutator between the operators that represent the measurements in question.
One might try to explain the second situation (the nonreversibility / probabilistic nature of the outcomes) by quantum hidden variables and try to argue that at a deep level the two effects are not really different and their difference only arises from ignorance. But Bell's Theorem rules even this out, if we are to keep a notion of locality in physics (i.e. a notion of relativistic, Lorentz-invariant notion of causality).
Hence, whereas the observer effect can be minimized in principle to be as small as one likes with enough experimental design effort, there is in principle no way to do outdo the constraints imposed by Heisenberg uncertainty.