Actually the moon is not white, it’s dark gray. The albedo (reflectivity) is only around 7%, IIRC. It looks so bright because it’s in full sunlight, and you usually see it at night when nothing else in sight is lit by sunlight.
As for the OP, are you talking about a disco mirror ball which is made up of numerous flat surfaces, or a smooth sphere? If it were a shiny smooth metal sphere, it would look pretty much as you’d expect it - a shiny sphere. It’d be about 10 times as bright as the moon we have now. Not enough to make any measurable change to the climate. If you keep it in the same place and same mass, there will be no change to the tides.
If you covered the moon with numerous flat mirrorrs, and if the mirrors are fairly large (few miles or more), it would be a bit different. The reflection from this mirror ball will be numerous narrow beams of light. If one of these reflections hits you directly, it’d be almost as bright as the full sun. When it isn’t hitting you, the moon would look much darker than it does now. If each mirror is of order hundreds miles or larger, it will probably affect local weather patterns.