What is the purpose of a "vapor barrier" under a concrete floor, does it make the bottom weak?
This is a political football of a question. I will get the easy answer out of the way first. A vapor barrier does not make the bottom of the concrete weak. There are excellent curing conditions in the bottom section of a concrete slab and these tend to make the concrete stronger.
I suspect your question sprang from hearing someone say the bottom of a concrete slab was soft or plastic, when laid over a vapor barrier. This is true for a comparative short time. The comparison is to the top section of the concrete slab. The top section of a concrete slab generally stiffens faster than the bottom section. Both regions will harden, but at differing rates.
Continuing to answer your question in reverse order. The purpose of a vapor barrier is to eliminate rising dampness from the subgrade. Concrete is a stiff and structural material, it is also porous to water or water vapor. Compared to the atmosphere above a slab, the interior of a concrete slab is a high pressure area. Moisture vapor can travel through the interior pore system of the concrete and reaching the surface, condense into liquid water under the floor coverings.
Floor coverings do not perform well when underlain with a minute but persistent supply of liquid water. Carpet dampens, vinyl tile losses its bond, urethanes age due to the high pH of the rising water and epoxies can become cloudy in appearance. Waterproof vapor barriers are intended to prevent any rising dampness from the surface.