Some get the difference here and some don't, so I thought I'd put some numbers behind what the physics suggests is the appropriate approach to each. I will use my data as I trust my numbers, but there is other information out there that I will link to so that you need not take my word on anything.
I will use the difference between E and D tunings for each instance for the sake of simplicity.
Drop Down Tuning is a shift of the entire string set. This keeps the intervals between the strings the same but lowers the pitch. This is often referred to as standard tuning and is specified to by the pitch of the lowest string. This also means that to keep the same or similar feel at the new tuning you need to adjust all the gauges of your strings. Your tuning would go from E A D G to D G C F if you were to drop down a whole step and have the same relationships and patterns on your fretboard.
Example; standard E set (my numbers and recommendation - ymmv)
E .100 - 38.7, A .076 - 40.2, D .055 - 37.7, G .040 - 37.1
To shift to a Drop Down D (D standard) tuning and keep the tension and feel consistent a set looks like this;
D .112 - 37.9, G .084 - 38.3, C .061 - 36.9, F .046 - 38.8
Drop Tuning lowers the pitch of the lowest string only and changes the relationship to a 5th interval, maintaining the standard 4ths intervals of the higher register strings. To accomplish this the only adjustment necessary is increasing the gauge of the lowest string.
Standard E from the example above;
E .100 - 38.7, A .076 - 40.2, D .055 - 37.7, G .040 - 37.1
To shift to a Drop D tuning a set would look like;
D .112 - 37.9, A .076 - 40.2, D .055 - 37.7, G .040 - 37.1
Shifting a whole step is pretty straight forward, but it can get confusing quickly if you are pursuing C, G, A or any other alternate tuning, Dropped or not. There are a multitude of variations and resources that can assist.
I approach sets and intervals differently than most manufacturers, but if you compare data you will find consistency in weights and tensions between what I do and what other manufacturers make available; there are only so many ways to wind a string so you can apply the available data to anyone's string offerings.
My unit weight and tension data is here;
Circle K Tension Chart. D'Addario is gauge-consistent with what you will find from nearly every other manufacturer and can be applied to any manufacturer's gauges;
D'Addario Tension Chart
I hope the linked resources prove useful, and I hope I haven't muddied the waters.