Below are the definitions provided by the Wiktionary:
somit – therefore, hence, thus
deshalb – hence, therefore
Is there any difference between these two words in terms of their meaning or how they should be used?
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Deshalb stresses that what was just said causes whatever is now following.
Ich habe Höhenangst. Deshalb bin ich nicht geflogen.
Es hat einen lauten Schlag getan. Deshalb sind die Pferde durchgegangen.
Somit on the other hand, is a lot weaker. There is still some kind of underlying continuity in the sense of cause → effect, but it is either very weak or not obvious or not important.
Wir haben jetzt dreizehn Spieler im Verein. Somit können wir zwei Mannschaften aufs Eis stellen.
Wir haben unser Tagessoll erfüllt. Somit können wir in den Feierabend gehen.
Trying to think of examples I realised that they are harder to invent for somit. I am inclined to say that it is the less common word.
I know one case in which "deshalb" is preferred over "somit".
"deshalb" can also be used for "that's why ...":
(pointing on something as your argument) That's why I never ride an elephant! Deshalb reite ich niemals auf einem Elefanten!
"somit" would be wrong.