How Gr(2,7) and Gr(3,6) are related? Consider the two types of Grassmannians Gr(2,7) and Gr(3,6) having  their plucker embeddings in $\mathbb P^{20}$ and $\mathbb P^{19}$ respectivley. The first one is 10-dimensional and latter is 9-dimensional, so each having codimension 10. We can easily compute their defining equations and both of them are defined by  35 equations.
By using computer algebra system(such as Macaulay 2 or Magma) we can also compute their Hilbert series and so the Hilbert numerator. If we do the computation then both  varieties have basically same numerator, which suggests that both idels have the same(kind of) free resolution. 
This raises the question whether  Gr(3,6) is a linear section of Gr(2,7) or not.?   
 A: There is a theorem of Fujita saying that Grassmannian is never a hyperplane section of a smooth variety unless it is $P^n$ or $Gr(2,4)$.
A: I don't know whether the ideals have the same kind of free resolutions, but $Gr(3,6)$ is definitely not a hyperplane section of $Gr(2,7)$. Otherwise, their $H^{\leq 8}$ would be the same by the Lefschetz hyperplane theorem. However, $H^6(Gr(3,6))$ is 3-dimensional and is spanned by $c_1^3, c_1c_2$ and $c_3$, and $H^6(Gr(2,7))$ is 2-dimensional: it is spanned by $c_1^3$ and $c_1c_2$. Here $c_i$'s are the Chern classes of the respective tautological bundles.
A: EDIT: The following idea doesn't work, for kind of obvious reasons (see the comments).

As other answers have pointed out, the answer to the original question is no.  However, taking up the idea from David Speyer's comment, both $Gr(3,6)$ and the Schubert divisor $\Delta$ in $Gr(2,7)$ are linear sections of $Gr(3,8)$, by linear subspaces of the same dimension.  Specifically, as Schubert varieties inside $Gr(3,8)$, $$Gr(3,6) = \Omega_{(2,2,2)}$$ and $$\Delta = \Omega_{(5,1)}.$$  (I'm using notation where $\Omega_\lambda$ has codimension $|\lambda|$, for $\lambda$ a partition inside the $3 \times (8-3)$ rectangle.)  One checks that these two Schubert varieties are both defined by the vanishing of 36 Plücker coordinates (on $Gr(3,8)$).  
Taking any curve in the Grassmannian of codimension-36 subspaces inside ${\Bbb P}^N$, where $N = \binom{8}{3}-1$, connecting the two linear spaces cutting out $Gr(3,6)$ and $\Delta$, you should get a flat family having these two as fibers, explaining why they have the same Hilbert polynomial.
