What's the difference between the words jegliche und sämtliche?
I looked for the answer in the Duden dictionary, but all I found was that they are a synonym for all (nachdrücklich).
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For my feeling: sämtliche expresses that there is some set of things which is completely covered by some property while jegliche points more specifically at each thing in the set. (However, both meanings can be very similar or synonymous.)
This is supported by the link you posted where jegliches but not sämtliches is listed under "stärker vereinzelnd, die Einzelglieder einer Gesamtheit betrachtend; jeder, jedes, jegliches".
Also, sämtlich can stand on its own as in das gilt sämtlich für Modelle dieses Typs ("this is valid for all models of this type") while a similar construction is not possible with jeglich (at least I don't think so).
Sämtliche Werke means "the complete works" of some author while jegliche Werke is unusual and would only make sense in a context where there is a specific focus on each of the works such as in jegliche Werke weisen diese Schwäche auf ("each of the works show this deficit").
Furthermore, it has been pointed out before that jegliche is often used in a negative context (which is not a rule for sämtliche):
often used in the negative: Der Sprecher enthielt sich jeglichen Kommentars. Jegliche Hilfe kam zu spät. The "no exceptions whatsoever" aspect is highlighted. Can be safely replaced by "alle" or "jeder/-r/-s", as a rule.
Here it is highly instructive to look at the origin of "jeglich" and some examples of usage:
mittelhochdeutsch ieclich, iegelich, althochdeutsch iogilīh, zusammengerückt aus: io, eo = immer (je) und gilīh = gleich (welcher), jeder. Duden
Hence, the original meaning is "no matter which", "any".
Please note the example Duden provides:
frei von jeglichem neidischen Gefühl.
I would translate this as "free of any envious feeling". "Sämtlich" can not be used in this example. The same is true of the following sentence:
«Er verschlang Bücher jeglicher Art.»
which I translate as "He devoured books of any kind".
"Sämtliche", on the other hand, is related to words like "Gesamtheit". It expresses "every". As in:
«Auf seine telephonische Verständigung hin wären innerhalb einer einzigen Stunde sämtliche Hotels und Pensionen von Paris auf das genaueste untersucht worden...» Stefan Zweig, Die Welt von gestern
We would not use "jegliche"'here.