to give up / to do without
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to give up
To voluntarily decide not to do, use, or claim something, even though one could or would like to.
Um gesünder zu leben, verzichte ich auf Zucker und Alkohol.
To live more healthily, I give up sugar and alcohol.
The verb 'verzichten' is almost always used with the preposition 'auf' followed by the accusative. You cannot use a direct accusative object without 'auf'.
Ich verzichte auf den Nachtisch, ich bin schon satt.
I'll skip dessert; I'm already full.
People often give up stimulants, habits, or possessions. Typical combinations are 'auf Fleisch verzichten' (to give up meat), 'auf das Auto verzichten' (to do without the car), or 'auf Süßigkeiten verzichten' (to give up sweets).
Viele Menschen verzichten in der Fastenzeit auf Alkohol.
Many people give up alcohol during Lent.
When you give up doing something (that is, an action), you use the construction 'darauf verzichten, zu + infinitive' or 'darauf verzichten, dass ...'.
Ich verzichte darauf, ihn heute noch anzurufen.
I won't call him today after all.
to use / benefit
'nutzen' and 'verzichten' are opposites here. When you 'nutzen' something, you use it or take advantage of an opportunity. When you 'verzichten auf' something, you consciously decide against it, even though it would be possible. 'nutzen' usually takes a direct object, for example 'den Bus nutzen' (to use the bus). 'verzichten' is used with 'auf', for example 'auf den Bus verzichten' (to do without the bus). So the contrast is: to use something or to deliberately not use it.
Nach dem Essen verzichtet sie heute auf Kaffee.
After the meal, she is skipping coffee today.
Bei Regen nutze ich lieber den Bus statt das Fahrrad.
When it rains, I prefer to use the bus instead of my bike.