When I added a resistor to a set of christmas lights where I cut off bulbs, it gets hot. Why does a shorter string of lights not need a resistor?

The strings are designed to use bulbs whose voltages sum to equal the supply voltage. So a string that uses 20 bulbs for a 120V power source will use bulbs designed to operate at 6 volts. And a string that uses 50 bulbs for a 120V source will use bulbs designed to operate at 2.4V. When you are making hundreds of thousands (or millions) of strings you can have custom bulbs made for whatever voltage you wish.

If you want to remove some of the bulbs and shorten the string, then you must compensate for the power the bulbs were using or run the risk of premature failure of the remaining bulbs which you are operating over-voltage.

A 2.4V bulb designed for a 50-bulb string is NOT "interchangeable" with a 6V bulb designed for a 20-bulb string. No matter how similar they may appear to the naked eye.

If you want to shorten a string of 100 bulbs to 78 (actually two parallel strings of 39 for 120VAC) you can wire a 1N4007 rectifier diode in series with the strings. It will reduce the RMS voltage by about 30%, which is about right.

Bulbs can be manufactured to any desired voltage.

I'd assume the strings use different voltage bulbs. Did you measure the voltage across each bulb when it's in it's string and lit?

Incandescent bulbs also don't have a rigid threshold of "working" vs "not working". The bulbs in one string might be overdriven with a voltage higher then they're really supposed to be run at, and the other string might be under driving the bulbs. Overdriving will produce more light, but shorten the lifetime. Underdriving will do the opposite, but you'll still get some light unless it's very, very underdriven.