
weiterlaufen
to continue walking
To continue moving on foot without stopping or taking a break.
Obwohl seine Füße schmerzten, ist er einfach weitergelaufen.
Although his feet hurt, he simply continued walking.
to continue running / to keep going
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to continue walking
To continue moving on foot without stopping or taking a break.
Obwohl seine Füße schmerzten, ist er einfach weitergelaufen.
Although his feet hurt, he simply continued walking.

to keep running
To be continued or remain in operation. Refers to contracts, processes, devices, or events.
Der Mietvertrag läuft automatisch weiter, wenn er nicht gekündigt wird.
The rental contract keeps running automatically if it is not terminated.

to keep showing
Of a film, broadcast, or performance: to continue being in the program or being shown.
Der Film war so erfolgreich, dass er noch drei Wochen länger im Kino weiterlief.
The film was so successful that it kept showing in cinemas for another three weeks.
'Weiterlaufen' (to continue running / to keep going) is a separable verb. The prefix 'weiter' comes at the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Film läuft noch eine Woche weiter.
The film keeps showing for another week.
The perfect tense of 'weiterlaufen' (to continue running / to keep going) is always formed with the auxiliary verb 'sein'. This applies both to literal movement and to the figurative meaning (e.g., with contracts).
Wir sind bis zum nächsten Dorf weitergelaufen.
We continued walking as far as the next village.
The prefix 'weiter-' means 'to continue' or 'to do continuously'. It can be combined with many verbs to create this meaning, for example in 'weiterarbeiten' (to continue working), 'weiterlesen' (to continue reading), or 'weitersprechen' (to continue speaking).
Nach der kurzen Pause lief die Musik einfach weiter.
After the short break, the music simply kept going.
You can recognize the meaning from the subject: If a person or animal is the subject, it is usually about walking. If a contract, device, process, or film is the subject, it means 'to be continued' or 'to remain in operation'.
Die Wanderer liefen schweigend weiter.
The hikers continued walking in silence.
In the figurative meaning, 'weiterlaufen' (to keep running) is often used for abstract or technical things. Typical subjects are: der Vertrag (the contract), das Abonnement (the subscription), die Kosten (the costs), die Produktion (production), or der Motor (the engine).
Solange Sie nicht kündigen, laufen die Kosten für das Abo weiter.
As long as you don't cancel, the costs for the subscription keep running.
to continue / move on
Both words can mean that someone is continuing to move on foot. Weitergehen clearly refers to continuing to walk. Weiterlaufen can also mean moving on in general, but depending on the context it can also refer to running faster; the immediate context determines this nuance.

Nach der Pause laufen die Jogger in schnellem Tempo weiter.
After the break, the joggers continue running at a fast pace.
Nach der Pause gehen die Wanderer in ruhigem Tempo weiter.
After the break, the hikers continue walking at a leisurely pace.