To begin, let's start with a very apropos anecdote from Lorentz himself. At a conference on the Michelson–Morley experiment in 1927 at which Lorentz and Michelson were present, Michelson suggested that Lorentz was the initiator of the theory of relativity. Lorentz then replied: “I considered my time transformation only as a heuristic working hypothesis. So the theory of relativity is really solely Einstein's work. And there can be no doubt that he would have conceived it even if the work of all his predecessors in the theory of this field had not been done at all. His work is in this respect independent of the previous theories.” Lorentz, replying to Michelson at the Solvay Conference.
Poincaré was confused on several points. (See the discussion on Wikipedia regarding “mass energy equivalence”.) He could never get the mechanical relations straight, since he could not figure out that $E=mc^2$. Einstein was aware of Poincaré's work, but he derived the theory simply as a geometric symmetry, and made a complete system from first principles using one, and only postulate: the speed of light is invariant and is the same for all frames of reference.
Einstein did share the credit with Lorentz and Poincaré for special relativity for a while, and Fitzgerald and Maxwell deserve credit for their contributions as well. However, it was Einstein, and Einstein alone, that was able to derive SR from first principles without ad hoc solutions to the transformation problem. The great physicist Wolfgang Pauli in the Encyclopædia Britannica article famously credits Einstein alone for formulating the relativity principle, as did Lorentz. Poincaré was less accommodating but that is probably because he never fully understood Einsteinian relativity and was mired in the Galilean conception of relativity in which there is always a privileged frame of reference. Both Lorentz and Poincare said some variation of “Einstein just assumed that which we were all trying to prove” (namely the principle of relativity) - Lorentz meant it affectionately whereas Poincare mean it as an insult. Poincaré was still trying to get relativity from Maxwell's equations, rather than making a new postulate, and this is a huge difference. See below.
Special relativity was ripe for discovery in 1905, but it was Einstein, and Einstein alone, who derived a complete system from first principles (using one fundamental postulate) and only he correctly derived $E=mc^2$ from said postulate. Without his derivation of $E=mc^2$, nothing makes sense. Poincaré, Lorentz, Fitzgerald, and Maxwell deserve 50% of the credit (as Einstein himself accepted), but that's like splitting hairs and pointing out that Descartes and Fermat deserve 70% of the credit for calculus when Newton and Leibniz were the first to integrate the various derivations of analytical geometry.
Lorentz actually misinterpreted his own transformations as applying to the ether (in fact I think he derived them purposely to describe how the ether “reacts” in such a way as to be in accord with experiments (Michelson-Morley). So Einstein gets credit for being the first to correctly interpret the equations and do away with the ether concept. Both Poincare and Lorentz believed in the ether - a non-trivial fact - which represents a privileged frame of reference (Galilean relativity) as opposed to a coordinate system that has NO preferred reference frame (Einsteinian relativity).
Einstein derived $E=mc^2$ from first principle, notably that energy carries inertia from emitter to absorber, and the resulting rest frame that the separation “mass” and “energy” entails. This derivation of $E=mc^2$ as a consequence of special relativity is epistemically critical to Dirac's later reconciliation of special relativity and quantum mechanics.
Einstein had the correct electrodynamic transformations in 1905, and the correct energy density and momentum density expressions, and the right relation between mass and energy. Poincaré had all but the last, and this got him confused on several important points. Poincaré did not have spacetime geometry before Minkowski because: A) he did not define it through non-galilean relativity, B) Poincaré did not express particle motion in terms of a worldline, or define proper time as a worldline parameter. So inasmuch as spacetime geometry includes worldlines and proper time, Poincaré did not discover spacetime geometry. Einstein's complete systemization of the Lorentz transformations and the second order transformations essentially wove space and time together in a way that is non-Galilean (Poincaré never got that far). Minkowski, took Einstein's interval relation and mathematized it in 4-d vectors (but again, it was already implied in Einstein's 1905 paper).
Poincaré noted a preference against using Minkowski spacetime. Another curiosa, Moskowzki sat in the audience at his Berlin lecture where Poincaré had talked about Einstein's work both in the positive and negative. The speech-transcript does not include any explicit mention of Einstein by name though, so we are left to assume he did a bit of adlib when talking about Einstein specifically. From Moskowzki's impression, it seems Poincaré did indeed regard Einstein's work as not only different to his but… too revolutionary and daring. See this link.
Einstein argued that a light pulse which is spherical in one inertial frame, is spherical in every inertial frame. According to Poincaré, a light pulse that is spherical in the above mentioned privileged frame is an elongated ellipsoid in every other inertial frame. The difference in description is due to that fact that Einstein recognized the relativity of spatio-temporal coordinates, when Poincaré did not. And, the aberration constant, Poincaré didn't derive it, Einstein did.
Although Poincaré understood independently of Einstein how the Lorentz transformations give rise to non-Galilean transformation rules for velocities (indeed Poincaré derived the correct relativistic rules), it is clear that he did not have a full appreciation of the modern operational significance attached to coordinate transformations. He did not seem to understand the role played by the second-order terms in the transformation. Compared with the cases of Lorentz and Larmor, it is clear that Poincaré did not fully understand either length contraction or time dilation to be a consequence of the coordinate transformation. What Poincaré was holding out for was no less than a new theory of ether and matter — something far more far-fetched than what appeared in Einstein's 1905 relativity paper. Einstein's rejection of the aether is important, but is not the pivotal difference between his conception of relativity and Lorentz/Poincaré conceptions. Why? Because it is always possible to add for whatever reason the notion of a privileged frame to special relativity, as long as one accepts that it will remain unobservable. However, in addition to the examples given above, there are other new features in Einstein's work.
*The full meaning of relativistic kinematics was simply not properly understood before Einstein. Nor was the 'theory of relativity' as Einstein articulated it in 1905 anticipated even in its programmatic form. It is impossible to understand the full implications of Einstein's discovery of special relativity without taking on board the impacts of the quantum in physics (see Paul Dirac). In respect to the conventional nature of distant simultaneity, Einstein was doing little more than expanding on a theme that Poincaré had already introduced. Where Einstein goes well beyond the great mathematician is in his treatment of the coordinate transformations. In particular, the extraction of the phenomena of length contraction and time dilation directly from the Lorentz transformations in section 4 of the 1905 paper is completely original. The genius of Einstein's 1905 paper is that the modern, dynamical interpretation of special relativity — as opposed to the kinematical approach of Einstein's 1905 paper — is ALREADY contained in Einstein's 1905-paper “masqueraded in the language of kinematics” (Physicist Harvey Brown) and the modern understanding of space-time.
You cannot formulate General Relativity using Poincaré/Lorentz's conceptualization of relativity. You can, and Einstein did, using Einsteinian SR. Differences, for instance, in how one obtains the conservation of mass in GR vary widely depending on whether you use Einstein's approach or Poincare/Lorentz. I hope this settles the debate once and for all. Until his death, Poicaré never fully grasped the full implications of what Einstein had done. Lorentz, a man Einstein called “the smartest man I have ever known,” eventually came around to it once General Relativity proved so successful.