
besonders
especially
As an adverb: to a high degree; especially, above all.
Der Film war gut, besonders das Ende hat mir gefallen.
The film was good; I especially liked the ending.
especially / special
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especially
As an adverb: to a high degree; especially, above all.
Der Film war gut, besonders das Ende hat mir gefallen.
The film was good; I especially liked the ending.

special
As an adjective: unusual, special, not everyday.
Heute ist ein ganz besonderer Tag für uns.
Today is a very special day for us.

especially
As an adverb: separately, apart, extra.
Ich möchte mich bei Ihnen noch einmal ganz besonders für Ihre Hilfe bedanken.
I would like to thank you once again especially for your help.
'Besonders' can be an adverb ('especially') or an adjective ('special'). As an adverb, it is not declined. As an adjective, it stands before a noun and must be declined.
Das Wetter ist heute besonders schön.
The weather is especially nice today.
As an adverb meaning 'vor allem' (above all) or 'speziell' (especially/specifically), 'besonders' usually stands directly before the word or phrase it emphasizes.
Ich mag Obst, besonders Erdbeeren.
I like fruit, especially strawberries.
As an adjective, 'besonders' (special) often describes things that are unique or important for an occasion. Typical combinations are 'ein besonderer Anlass' (a special occasion), 'ein besonderes Geschenk' (a special gift), or 'eine besondere Ehre' (a special honor).
Zu meinem Geburtstag habe ich ein besonderes Geschenk bekommen.
For my birthday, I received a special gift.
own / peculiar
Besonders describes an atmosphere as unusual or remarkable. Eigen, on the other hand, emphasizes that the atmosphere is characteristic of that particular place. An atmosphere can therefore be both besonders and eigen, depending on whether its effect or its distinctive character is meant.

Das kleine Café hat eine besonders angenehme Atmosphäre.
The small café has an especially pleasant atmosphere.
Das kleine Café hat eine ganz eigene Atmosphäre.
The small café has a distinctive atmosphere all its own.