
aufwärts
upward
Describes a movement or direction from a lower point to a higher point.
Wir stiegen den steilen Pfad langsam aufwärts.
We slowly climbed upward along the steep path.
upward
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upward
Describes a movement or direction from a lower point to a higher point.
Wir stiegen den steilen Pfad langsam aufwärts.
We slowly climbed upward along the steep path.

upward / improving
In a figurative sense, a positive development or improvement of a situation, e.g. in the economy or one's personal mood.
Nach der Krise geht es mit der Wirtschaft endlich wieder aufwärts.
After the crisis, the economy is finally improving again.
The word consists of 'auf' (up) and the directional suffix '-wärts'. You find this suffix in many directional adverbs such as 'abwärts' (downward), 'vorwärts' (forward), or 'rückwärts' (backward).
Der Ballon stieg langsam aufwärts in den Himmel.
The balloon slowly rose upward into the sky.
You can almost always recognize the figurative meaning from the fixed expression 'es geht aufwärts' (things are improving). If, on the other hand, it refers to concrete movement, it is often combined with verbs such as 'gehen' (to go), 'steigen' (to climb/rise), or 'fahren' (to drive/travel).
Wir müssen den Berg noch zwei Kilometer aufwärts gehen.
We still have to walk two kilometers upward on the mountain.
To express a positive development, the impersonal expression 'es geht aufwärts' (things are improving) is almost always used. You can also say what is improving by adding 'mit' + dative.
Nach langer Krankheit geht es mit ihm endlich wieder aufwärts.
After a long illness, he is finally improving again.
up
Rauf is colloquial and describes movement upward, often toward a named higher place. Aufwärts emphasizes the direction or the upward course more strongly. Both are possible for many concrete movements, but the perspective is slightly different.

Vom Eingang führt die Treppe steil aufwärts zum Aussichtsturm.
The stairs lead steeply upward from the entrance to the observation tower.
Wir gehen jetzt rauf auf den Aussichtsturm.
We're going up to the observation tower now.