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.

'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 run
'laufen' and 'weiterlaufen' can both describe movement, operation, or something ongoing. The most important difference is: 'weiterlaufen' emphasizes that something does not stop, but continues. A person can run or walk; if they do not stop after a break or despite an obstacle, they 'läuft weiter' (keep going / continue running or walking). A device, a contract, or a film can also 'laufen' (run / be running / be showing); with 'weiterlaufen', you say more precisely that it continues to run. The words can be confused because 'laufen' already describes an active or ongoing situation. When continuation is important, 'weiterlaufen' fits better.

Trotz des Stromausfalls läuft der Server mit Batterie weiter.
Despite the power outage, the server keeps running on battery power.
Der Drucker läuft wieder, nachdem Tom das Kabel geprüft hat.
The printer is running again after Tom checked the cable.