When you perform the measurement on which slit the particle passes through, then the measuring device is also part of the system and it affects it. The measurement reduces the degrees of freedom in the system so there are no longer two equivalent ways for the particle to pass through the slits (either A or B), but rather you now have a measured slit and an unmeasured slit. Since there are no longer multiple ways to achieve the same result, the is no longer interference due to equivalent probabilities.

Matt Stassler has a nice series of blog posts on this.