Depending on your SVM implementation, there may be a difference.
Compared to the linear kernel, the polynomial kernel has an additional parameter $c$ (and $d$ of course):
$K(x,y) = (x^\mathsf{T} y + c)^{d}$
However, if $c=0$ (and $d=1$), then:
(linear) $K(x,y) = x^\mathsf{T} y \equiv K(x,y) = (x^\mathsf{T} y + c)^{d}$ (polynomial)
Nonetheless, some SVM implementations may opt-out of calculating $c$ and assume homogeneity in order to reduce the number of hyper-parameters. So, it depends on how your SVM is implemented.
For example sklearn's polynomial kernel assumes by default $c=1$ if you do not specify the parameter, which would make it not exactly the same as a linear kernel.