Known problems from reduced gravity:
- Motor muscle atrophy
- Heart muscle atrophy
The natural process of bone replacement is compromised in 0-G. It's also been shown by experiments to be compromised by lack of long-axis stress on the long bones - stresses like walking induce microfractures which proceed to get "healed" with stronger bone. How much gravity is needed to maintain bone health is not yet known, but the lack of long-bone stress can be compensated by vigorous exercise with elastic bands holding one down. (This has been demonstrated on US space stations.)
Motor muscle atrophy can be partially compensated by vigorous exercise with elastic resistance. This is not sufficient to maintain full pre-flight strength for those in microgravity.
Peripheral cardiovascular effects are apparently temporary (cite). The heart muscle, like all muscle, will not maintain condition and muscle density when loading is reduced. Lower gravity will require less pressure to push blood up into the brain; this reduced effort will result in lowered heart muscle condition, and eventually, atrophy. (cite)
The use of a weighted vest will increase both required muscle effort for walking, and impact stress from walking. The decreased velocity, however, will still not result in the same effect as normal gravity, at least not without excessive mass. (Remember, E=MV^2.)
The use of a weighted vest will not, however, result in increased direct load on the cardiovascular system, so heart atrophy is likely.
TLDR: No, a weighted suit will not eliminate the health effects of reduced gravity. It will ameliorate some of them.