How to explain the selectivity between methanol and methanethiol in an SN1 reaction with an halogenated hydrocarbon?
Obviously saying "the rate is same but the activation barrier is different" is a contradiction, as both are directly related via the arrhenius equation. The rate is predominantly determined by the rate determining step. If there are several rate determining steps (aka several steps with similar activation energies), the overall rate will be a product of all steps.

So what does that mean for your question?

You stated that methanethiol is a better nucleophile than methanol. This would suggest that the activation barrier in the last step is slightly lower in the case of methanethiol. However, the rate determining first step is unaffected. Hence, the overall rate will be only slighter smaller for methanol. 

The ratio of both products will be determined by the last step.