morgen
tomorrow
Refers to the day that follows today.
Morgen habe ich einen wichtigen Termin.
Tomorrow I have an important appointment.

tomorrow / morning
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tomorrow
Refers to the day that follows today.
Morgen habe ich einen wichtigen Termin.
Tomorrow I have an important appointment.

The spelling changes the meaning: the adverb 'morgen' (written with a lowercase letter) means 'the next day'. The noun 'der Morgen' (written with a capital letter) means 'the morning'.
Wir fahren morgen in den Urlaub.
We're going on vacation tomorrow.
You can combine the two meanings of 'morgen' (tomorrow/morning). The expression 'morgen Morgen' is correct and means 'in the morning of the next day'.
Unser Flug geht morgen Morgen um 6 Uhr.
Our flight leaves tomorrow morning at 6 o'clock.
To indicate a specific time on the next day, you combine 'morgen' (tomorrow) with a time of day, for example 'morgen früh' (tomorrow morning), 'morgen Mittag' (tomorrow noon), or 'morgen Abend' (tomorrow evening).
Können wir uns morgen Abend im Restaurant treffen?
Can we meet tomorrow evening at the restaurant?
The single word 'morgen' almost always means 'the next day'. To express 'in the morning', people usually use 'am Morgen' (in the morning) or combinations like 'heute Morgen' (this morning) or 'gestern Morgen' (yesterday morning).
Heute Morgen habe ich die Nachrichten gelesen.
This morning I read the news.
In the meaning 'future', 'morgen' is often used in a somewhat elevated or philosophical context. The combination with 'von' is typical: 'die Welt von morgen' (the world of tomorrow).
Wir müssen heute die Grundlagen für die Gesellschaft von morgen schaffen.
We must create the foundations for the society of tomorrow today.
today
'heute' and 'morgen' both refer to days close to the moment of speaking. 'heute' means the day on which you are speaking. 'morgen' means the next day after that. The words are easily confused because both often appear with appointments, plans, and everyday situations. The practical difference is simple: what happens on the current day is 'heute'. What happens on the next day is 'morgen'.

Wegen des Wetters trifft meine Kollegin mich morgen nicht im Park, sondern im Büro.
Because of the weather, my colleague is meeting me tomorrow not in the park, but at the office.
Er besucht heute seine Eltern vor dem Termin.
He is visiting his parents today before the appointment.