problem with Simple Timer program

OK. I changed the code you posted as follows:

Pin changed from 8 to 13 for onboard LED.
12 hour timer reduced to 4000 ms for testing.
5 minute delay reduced to 1000 ms for testing.

#include <SimpleTimer.h>   // from the simple timer library

SimpleTimer timer;
void RepeatTask() {
  digitalWrite(13,HIGH);  //this connects to LED

  delay(1000);         // wait 1 second
}

void setup() {
  Serial.begin(9600);
  pinMode(13,OUTPUT); // LED
  
  // timed actions setup
  timer.setInterval(4000, RepeatTask); // for LED. repeats every 4 seconds
 }

void loop() {
    timer.run();  // from simple timer example.
}

This code ran identically with Arduino Uno connected to PC and connected to external power supply: after power on or reset, LED flashes briefly (Arduino booting up), LED goes off, LED comes on after 4 seconds, then stays on indefinitely (since there is no digitalWrite(13,LOW) to turn it off.

So the code does what I would expect, both with USB power and external power.