The kernel you choose defines the function class you're working with.

The squared exponential kernel defines a function space that is a lot larger than that of the linear kernel or the polynomial kernel.

A linear kernel allows you to use linear functions, which are really impoverished. As you increase the order of the polynomial kernel, the size of the function class increases. An n-th order polynomial kernel gives you all analytic functions whose derivatives of order (n+1) are constant, and hence all derivatives of and above order (n+2) are zero. The squared exponential kernel gives you access to all analytic functions (that is all infinitely differentiable functions). So in some sense you can view the SE kernel as being as powerful as an infinite order polynomial kernel. (this is technically incorrect, but a useful way to start understanding the differences)

Technically if you use squared exponential kernel, than you're method is nonparametric, if you use polynomial kernels, you're model is parametric. In a way nonparametric model means that the complexity of the model is potentially infinite, it's complexity can grow with the data. If you give it more and more data, it will be able to represent more and more complex relationships. In contrast a parametric model's size is fixed, so after a certain point your model will be saturated, and giving it more and more data won't help. So asymptotically, assuming you have unlimited data and very weak assumptions about the problem, a nonparametric method is always better.

In conclusion, the squared exponential kernel is generally more flexible than the linear or polynomial kernels in that you can model a whole lot more functions with its function space.

However, if you know (i.e. it is reasonable to assume) that your true function can be well approximated by a linear or polynomial function, then a linear or polynomial kernel will be much better, it will fit that polynomial for you using a lot less data then the squared exponential.