It should be said that Husserl was philosophically averse to Kant's "creative" transcendental subject, perhaps due to the dominance of absolute idealist interpretations of him at the time, and preferred to derive his lineage from Hume, whom he credits as the principal forerunner of phenomenology. See Mall's Experience and Reason on their connection, which quotes Husserls' 1919 letter to Metzger:"I have learnt incomparably more from Hume than from Kant. I possessed the deepest antipathy against Kant, and he has not (if I judge rightly) influenced me at all".
This is likely exaggerated, Kant was in the air of the times, and Husserl did draw parallels with him already in Logical Investigations (1901), while reinterpreting his "synthetic" notion of a priori into intuitive one, and adopted the label and terminology of transcendental idealism around 1915, apparently at Natorp's prompting. There are also undeniable parallels between his approach and that of contemporary neo-Kantians, especially in the early works. The Crisis, on the other hand, is a late work, not completed in Husserl's lifetime, and written under pronounced influence of existentialism, in particular Heidegger's. One has to keep in mind though that what appears to be Kantian framework in Husserl may be at least partly attributable to their common root in Hume. I recently came across a very nice summary of Husserl's theory of cognition in Zhok's Ontological Status of Essences in Husserl’s Thought, which makes Husserl's affinities and breaks with Kant more transparent.
What Husserl kept and developed beyond Hume was the recognition of the creative role of mind in cognition. Productive imagination (term shared with Kant) plays a dual role of aiding apprehension in perception, producing perceptual unities like objects, and engages in "free play" to produce what Kant called synthetic a priori intuitions in arithmetic and geometry. In fact, Husserl expands its role even further, it is responsible for memory recalls and "eidetic variation" of acquired perceptual proto-concepts that sharpens their boundaries, and forges them into full fledged essences ("eidoses"). As a result, Husserl gives a more satisfying account of empirical concept formation, which was a major unresolved problem for Kant, see Pippin's Kant on Empirical Concepts. He could not explain how exactly definitive concepts are formed from the undifferentiated "manifold of sensation", and his German idealist successors turned it into "construction of reality" from mental categories.
And here we come to a major break with the Kantian theory of sensibility, Husserl rejects the undifferentiated manifold, and the idea that percepts are synthesized from "sense data". The latter is seen as ex post facto extraction from what is originally given to consciousness as already partially structured and unified, if obscurely (this was later confirmed by empirical cognitive science). In other words, rather than having two determinates, "sensibility" and "manifold" interacting, Husserl insists that all determinacy is only forged in the act of perception itsel. No determinates can be presented as "interacting" prior to it, dissolving the question of whether the content of perception "pre-exists" in reality, or is generated by mind.
This aspect of perception, which apprehends idealities as immediate unities, Husserl terms categorical intuition, and together with eidetic variation, which shapes concepts into definitional maturity, it forms the process of ideation. Thus, Husserl sails between the Scylla of passive reception of impressions à la Hume, and the Charybdis of German idealist "construction of reality" by the mind.
Categorical intuition releases Husserl from the need to keep perhaps the most implausible part of Kantian picture, the forever immutable a priori categories and forms of intuition. But it is not the intellectual intuition of Spinoza and Fichte, it captures invariances of sensuous experience, not "glimpses" of things in themselves. But with it Husserl can be more generous on what is almost a vanishing point in Kant, the unknowable X. The transcendence, as Husserl calls it, is that content of consciousness that "points beyond" consciousness itself, given to it as not its own but foreign, subject to pre-cognitive awareness as the "raw matter" of sensuous experience. The limited creative ability to perceive wholes, however weak and partial, ability that Kant denied us completely as "intellectus archetypus", allowed Husserl to remove some of the other-worldliness surrounding Kantian "supersensible substrate of experience", although as for Kant it remains beyond the reach of knowledge.
There are many other divergences, I'll mention one of particular interest to me. In the Second Analogy Kant gives a notorious transcendental argument for the a priori status of strict causality as condition of the possibility of our forming temporal succession of events (since they come with no time labels attached). Husserl's analysis of perception shows instead that the "now", like "sense data", is an ex post facto abstraction. In perception we instead encounter the "specious present" (the idea likely coming from James, along with the "stream of consciousness"), a short but dynamic duration with markers that explicitly link it to neighboring durations in the time succession, like coming and going notes in the apprehension of a melody (this was also confirmed by empirical psychology). Thus Husserl again grants us an immediately holistic grasp, however obscure and fleeting, this time of becoming. This removes another issue that caused Kant much grief (in his theory of free will), the necessity of unbreakable causal chains.
