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.

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' and 'speziell' can both highlight something. 'besonders' often sounds more general: something is beautiful, important, or unusual to a high degree. 'speziell' fits better when something is intended exactly for a particular purpose, person, or situation. As adverbs, both can also mean 'above all'. Learners can easily confuse the words because they seem similar in some sentences. A simple guide is: 'besonders' emphasizes the degree or special quality, while 'speziell' emphasizes the exact selection or the specific purpose.

Für Kinder gibt es in der Praxis einen speziell eingerichteten Warteraum.
At the clinic, there is a specially arranged waiting room for children.
Der Kuchen schmeckt heute besonders gut.
The cake tastes especially good today.