Detecting if one voltage is bigger than another without an operation amplifier (OP-AMP)
Assuming you only have access to basic components such as resistors and transistors, a very effective solution here is to use two transistors configured as a long tailed pair: See Wikipedia Description

In fact this is the input stage of almost all op-amps 

(Picture Source: Wikipedia as referenced above).



An excellent description of how to calculate gains and component values by Analog Devices Here

To get a "Digital Output" just use one of the outputs. The gain should be high enough that the output switches nicely between high and low.