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Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. In dimension 4 (where the J-curves live) we have the adjunction formula.

3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

Here is another crucial point disguised as an application: On a symplectic 4-manifold with (negative) contact 3-manifold boundary, the standard "ECH curve count" yields a relative invariant in the $ECH_*$ of the boundary, while the standard "SFT curve count" yields a relative in variant in the (ordinary) contact homology $CH_*$ of the boundary. But if the contact form is overtwisted then $CH_*$ is necessarily trivial, whereas $ECH_*$ can easily be nontrivial.

Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. In dimension 4 (where the J-curves live) we have the adjunction formula.

3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

Here is another crucial point disguised as an application: On a symplectic 4-manifold with (negative) contact 3-manifold boundary, the standard "ECH curve count" yields a relative invariant in the $ECH_*$ of the boundary, while the standard "SFT curve count" yields a relative invariant in the (ordinary) contact homology $CH_*$ of the boundary. But if the contact structure is overtwisted then $CH_*$ is necessarily trivial, whereas $ECH_*$ can easily be nontrivial.

Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. In dimension 4 (where the J-curves live) we have the adjunction formula.

3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. In dimension 4 (where the J-curves live) we have the adjunction formula.

3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

Here is another crucial point disguised as an application: On a symplectic 4-manifold with (negative) contact 3-manifold boundary, the standard "ECH curve count" yields a relative invariant in the $ECH_*$ of the boundary, while the standard "SFT curve count" yields a relative in variant in the (ordinary) contact homology $CH_*$ of the boundary. But if the contact form is overtwisted then $CH_*$ is necessarily trivial, whereas $ECH_*$ can easily be nontrivial.

Without elaborating much there are three key points, with the first one giving reason for the second one to happen:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)

3. In dimension 4 (where the J-curves live) we have the adjunction formula.

Without elaborating much there are three key points, with the first two laying the bedrock for the third:

1. ECH counts J-curves without caring about most information of the actual branched covers of such curves. Relatedly and more to the point, ECH counts J-curves with certain ECH index, and this picks out "the right" curves (separating itself from SFT).

2. In dimension 4 (where the J-curves live) we have the adjunction formula.

3. A lot of results deal with nontriviality of ECH, which comes from nontriviality of monopole Floer homology. (Ex: nontriviality of monopole Floer is what Taubes used to get existence of Reeb orbits, i.e. proof of Weinstein conjecture.) This is for the same reason that the Seiberg-Witten invariants are so powerful, because Taubes' SW = Gr result gives nontriviality results about symplectic 4-manifolds. (ECH is the "categorification" of the Gromov invariants.)