It appears you have a stone containing fossils. The grey stuff would be fossil remains~ cleaning beyond this point will harm the material. Picking at it with a sharp scribe can probably free up some material and expose other fossils under the layers.
I wonder if the lightly acidic cleaning properties of Extra Virgin Olive Oil might loosen and help remove some of the heavier grey sediment??!! I have used it on Copper coins for many years to remove dirt, grime and encrustations and believe it would work on rocks as well. Even if it does not loosen and remove the sediment, it will probably stabilize the rock from decay and give it a nice appearance. If you decide to try Extra Virgin Olive Oil, afterward it's use, just rinse the rock with water then dry with a towel.
Both vinegar and lemon juice are used on coins depending what they need but I think that they could be harmful to certain types of rocks, especially vinegar. However, I am not totally sure but you could do a test on a very small section of rock where it would not be noticed or noticed as much in case it is harmful.
I would echo huntsman53's concerns about vinegar, I personally would not use vinegar or anything that acidic on a fossil specimen like that. Vinegar is great for cleaning off sediment like that but the fossil details could be eaten away in the process. Keep the vinegar around for a quartz/agate (for example) specimen that has sediment on it though, I have had excellent luck giving agates/quartz a vinegar bath to eliminate severe sediment.