The use of the word notwithstanding in contracts does not differ from its plain and ordinary English meaning. Notwithstanding means in spite of, despite, even if, without regard to or impediment by other things, all the same, however, in any case, in any event, nevertheless, none the less, still, yet. Like subject to, the word notwithstanding creates a priority of provisions.

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Note added at 57 mins (2020-07-17 22:35:34 GMT)
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The use of notwithstanding might result in (unintended) ambiguities; when a third contract clause refers to a section that is itself subordinate to another clause, which uses notwithstanding: in that case it might be uncertain whether the prevailing clause (i.e. in which the reference notwithstanding was made) is intended to be captured by such third contract clause. If the section referred to in the third contract clause is subordinated to another clause by using subject to, the ambiguity does not occur because in that case it is clear that the subordinated section is limited by that other clause as well (since the subordinated clause says so itself).

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Note added at 1 hr (2020-07-17 22:39:08 GMT)
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“Notwithstanding” looks back to the main rule.
It is used in a clause that is to take priority over
another clause. It tells the reader that the subject
clause overrides the clause(s) to which it refers.
It is, in effect, the opposite of “subject to”.


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Note added at 1 hr (2020-07-17 22:41:06 GMT)
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"without prejudice to"
This expression is used in a clause where
there is to be no priority given over another
clause. It tells the reader that the clause in
which the expression appears does not affect
the clause to which it refers.

If you want to keep it simple.

Notwithstanding anything to the contrary contained herein. Most written contracts have many moving parts. Sometimes at the end of negotiations, the parties agree on something that might vary from something else they previously agreed to. ... It means that the contract might say two different and inconsistent things.