Refinishing slate tiles with worn-out portions
I would start with a light acid etch or whatever you do to reseal and bring a shine out will magnify the surface scum/ effervescence. 
I use muriatic acid and water with this I would start out with a weak solution 30:1 with possibly a max of 15:1.

I would mix up a small batch remembering to always add acid to the water AAA to prevent a thermal reaction that happens if water is added to acid. 

Safety glasses and rubber gloves will be enough safety gear and a good idea to have the vent fan on or window open. I would sponge the areas staying away from the grout or not etching the grout as heavily. 
Once the surface looks good rinse the sponge and then rinse the etched area at least 3 times and make sure it is fully dry Pryor to coating with a quality sealer.

The reason I suggest trying not to etch the grout it may really change colors and then the entire floor would need to be etched so the color will match. 
Start with a weak solution and possibly 2 or 3 wipes of fresh solution prior to going stronger is the best advice I can provide. 

Muriatic acid can be found in pool supply stores and most lumber / big box stores as this is commonly used to balance the ph in pools and to etch concrete and stone products.