How can you tell if something happened with cause or without cause? In both cases something happened. And causes cannot be observed. So, why do we infer the existence of causes in the first place?

Be careful that you don't confuse "cause" with "pattern" or "regularity". I can observe patterns and regularities. I cannot observe causes.

6.362 What can be described can happen too: and what the law of
causality is meant to exclude cannot even be described.


6.363 The procedure of induction consists in accepting as true the
simplest law that can be reconciled with our experiences.


6.3631 This procedure, however, has no logical justification but only a
psychological one. It is clear that there are no grounds for believing
that the simplest eventuality will in fact be realized.


6.36311 It is an hypothesis that the sun will rise tomorrow: and this
means that we do not know whether it will rise.


6.37 There is no compulsion making one thing happen because another has
happened. The only necessity that exists is logical necessity.


6.371 The whole modern conception of the world is founded on the
illusion that the so-called laws of nature are the explanations of
natural phenomena.


6.372 Thus people today stop at the laws of nature, treating them as
something inviolable, just as God and Fate were treated in past ages.
And in fact both are right and both wrong: though the view of the
ancients is clearer in so far as they have a clear and acknowledged
terminus, while the modern system tries to make it look as if everything
were explained.

How can you tell if something happened with cause or without cause? In both cases something happened. And causes cannot be observed. So, why do we infer the existence of causes in the first place?