
speziell
special
Particular, specific, or intended for a single purpose; not general.
Für diese Schraube brauchen wir einen speziellen Schraubenzieher.
For this screw, we need a special screwdriver.
special / specifically
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special
Particular, specific, or intended for a single purpose; not general.
Für diese Schraube brauchen wir einen speziellen Schraubenzieher.
For this screw, we need a special screwdriver.

particular / odd
Unusual, peculiar, or strange in behavior or appearance.
Er hat einen sehr speziellen Humor, den nicht jeder versteht.
He has a very particular sense of humor that not everyone understands.

specifically / especially
In particular, above all. Used as an adverb to emphasize something.
Die neuen Regeln gelten für alle, aber speziell für die neuen Mitarbeiter.
The new rules apply to everyone, but specifically to the new employees.
The word 'speziell' (special/specific) is related to 'Spezies' (species) and 'Spezialist' (specialist). All these words have the basic idea of 'type' or 'kind' and describe something that belongs to a particular, special kind.
Ein Spezialist ist ein Arzt für ein spezielles medizinisches Gebiet.
A specialist is a doctor for a special field of medicine.
The meaning of 'speziell' depends strongly on context. If it refers to an object or purpose, it usually means 'special' or 'made specifically for that'. If it describes a person, their taste, or their behavior, it can also mean 'peculiar' or 'odd'.
Ich suche eine spezielle Software für meine Arbeit.
I'm looking for special software for my work.
When 'speziell' means 'for a specific purpose', it is often combined with factual nouns. Typical collocations are 'ein spezieller Fall' (a special case), 'ein spezielles Werkzeug' (a special tool), 'spezielle Kenntnisse' (special knowledge/skills), or 'eine spezielle Aufgabe' (a special task).
Für diese Reparatur benötigt man spezielle Kenntnisse.
For this repair, you need special skills.
In the meaning of 'insbesondere' (in particular), 'speziell' is used like an adverb to emphasize part of the sentence. It often stands directly before the word or phrase that is meant to be emphasized.
Ich mag Obst, speziell Erdbeeren.
I like fruit, especially strawberries.
especially / special
Besonders emphasizes that something is unusual or not ordinary. Speziell focuses more on something being intended for a particular purpose or area. An object can be both, but speziell is more precise when referring to its specific function.

Das ist ein spezielles Messer zum Schneiden von Brot.
That is a special knife for cutting bread.
Das ist ein besonders schönes Messer.
That is an especially beautiful knife.