
herzlich
warm / cordial
In a very friendly, warm, and open way that comes from the heart. Often used in greetings, invitations, or congratulations.
Herzlich willkommen in unserem neuen Haus!
A warm welcome to our new house!
warm / heartfelt
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warm / cordial
In a very friendly, warm, and open way that comes from the heart. Often used in greetings, invitations, or congratulations.
Herzlich willkommen in unserem neuen Haus!
A warm welcome to our new house!

heartfelt
Sincere and deeply felt. Often used to express sympathy or thanks.
Mein herzliches Beileid zum Tod Ihrer Mutter.
My heartfelt condolences on the death of your mother.

completely
Used colloquially and often ironically to strongly emphasize disinterest or indifference.
Ob er kommt oder nicht, ist mir herzlich egal.
Whether he comes or not, I couldn't care less.
The adjective 'herzlich' (warm/heartfelt) is derived from the noun 'das Herz' (the heart). It describes an action or feeling that figuratively 'comes from the heart' and is therefore perceived as sincere and warm.
Er begrüßte seine Gäste mit einer herzlichen Umarmung.
He greeted his guests with a warm hug.
'Herzlich' (warm/heartfelt) is very often used in fixed expressions for greetings, wishes, and thanks. Typical examples are 'herzlich willkommen' (a warm welcome), 'herzlichen Glückwunsch' (congratulations), or 'herzlichen Dank' (many heartfelt thanks).
Herzlichen Glückwunsch zum Geburtstag!
Happy birthday!
'Herzlich' (warmly/heartily) can also be used as an adverb to describe a verb. It then means 'in a warm/heartfelt way' and expresses a deep, honest emotion.
Nach dem Witz mussten alle herzlich lachen.
After the joke, everyone had to laugh heartily.
You can recognize the colloquial, ironic meaning from the context. Here, 'herzlich' (completely/utterly) usually appears together with words like 'egal' (doesn't matter) or 'gleichgültig' (indifferent) to strongly emphasize disinterest.
Was die anderen über mich denken, ist mir herzlich egal.
What others think about me couldn't matter less to me.
nice / dear
Liebe Grüße and herzliche Grüße both express warmth. Liebe Grüße usually presupposes a closer personal relationship. Herzliche Grüße also sounds warm and sincere, but can be addressed to people one is somewhat less close to.

Am Ende der Nachricht an ihre neue Kollegin schreibt Mia herzliche Grüße.
At the end of the message to her new colleague, Mia writes, “Warm regards.”
Am Ende der Nachricht an ihre beste Freundin schreibt Mia liebe Grüße.
At the end of the message to her best friend, Mia writes, “Kind regards.”