Each type of aircraft has different acceptible performance parameters, so it's difficult to give you a one-size-fits-all type of answer. If I can, what I'll do is give you a general example, which may then vary for a specific type of aircraft.
Snow (or slush) on a runway (commonly referred to as runway "clutter") affects both takeoff and landing performance. For takeoffs, clutter retards normal rates of acceleration, and for landings, braking (DEacceleration) distances are lengthened. The effect of both is that it makes a runway of X-length perform as if it's LESS than X-length, and since the airport folks can't come out and pour additional concrete to extend the runway on short notice
,aircraft weights must be reduced to keep the aircraft with safe allowable limits. Sometimes, this means that some folks get left behind even though there was an empty seat for them.
Generally speaking, an aircraft may not operate (that's takeoff or landing) on any runway covered by more than 1/2 inch of slush, 1 inch of wet snow, or 4 inches of dry snow. Anything beyond that, and the aircraft does not operate.
For takeoffs below the clutter levels mentioned above, the normal calculated runway takeoff weight is reduced by a factor of 5% or 10%, whether the clutter is minimal, or is closer to the max allowable values.
For landings, the clutter affects the "braking action" of the runway. A dry runway would (obviously) be the best for braking action, and is known as "dry" or "normal". As braking action deteriorates, the classifications range downward from "good", then "fair", then "poor", and finally, "NIL". Aircraft operations are prohibited on "NIL" surfaces, and it should also be noted that operations with "fair" and "poor" reported braking action also entail increasinly more restrictive crosswind limitations.
How is all this stuff measured? That's where things get interesting, as there are both OBjective and SUBjective methods in place.
The airport operator has the responsibility (here in the US, anyways) to have a snow control program in place, and they are generally pretty good about keeping everyone up-to-date of local field conditions. Some of this accomplished via issuance of a NOTAM (notice to airmen), but some stuff changes so quickly, that tower issues the information.
For takeoffs, snow depth is mostly commonly reported (objectively) by airport folks actually out there. A pilot can certainly make a depth determination, but obviously, doing so from from purely within the cockpit is a more subjective endeavor.
For landings, braking action can be assessed by either specially-equipped vehicles (that provide a numerical reading like MU, or other, that can be converted to good/fair/poor/NIL), or the pilots of any landing aircraft can report the braking action they experienced when they landing. Given the above two mechanisms, it's easy to see that one has more opportunity for subjectivity than the other.
Considering all the information above, who makes the decision? It sort of depends. If the airport operator decides that the clutter has exceeded limits and then closes the runway for plowing/brooming/de-icing, it's a no-brainer. That aside, here in the US, under FAR 121 Domestic/Flag rules, the captain and/or aircraft dispatcher make the call. If one of those two thinks they have limits, but the other has more current info that says they don't, the most conservative info takes precedence. I know there are probably some crews out there who will disagree, but I can assure you that within my 23-year career, I have personally told crews NOT to land on occasion because I had more timely info than they did. Does't happen often that way, but it does happen.
Sorry to be so long, but a complicated question...
ALL views, opinions expressed are mine ONLY and are NOT representative of those shared by Southwest Airlines Co.