Time complexity of Rabin-Karp algorithm
The running time you quote isn't correct. If $n = m$ then it takes $\Omega(n)$ to verify that $T = P$. The best case running time of any string matching algorithm is $\Omega(m)$, since this is how long it takes to verify a match. Perhaps you are disregarding the time it takes to compute the hashes.

Ignoring all of this, let us suppose that we are considering an algorithm whose running time is $\Theta(n-m+1)$, where $m \leq n$ are two parameters. As your example makes clear, this is not the same as $\Theta(n)$, since if $m$ is close to $n$ then $n-m+1$ is much smaller than $n$. Generally speaking, an asymptotic expression depending on two parameters cannot be reduced to an asymptotic expression depending on only one of them.

In practice, often $m$ is itself a known function of $n$, and this can be used to reduce $\Theta(n-m+1)$ to an asymptotic expression depending on a single parameter. For example, if $m = n/2$ then $\Theta(n-m+1) = \Theta(n)$, whereas if $m = n$, then $\Theta(n-m+1) = \Theta(1)$.