Since you have an onboard PWM module running in the background, you can also have a second pwm by software running in the foreground.

The code would be as follows.
You are now using two buttons, one is for LED and the other is for Fan.
I suggest that you use Fan on GPIO.2 (on board module), and LED on GPIO.5 pin.
Also, you have button on GPIO.3, and you are puling it down .
Nowyou have a second button but is is being pulled high.



Code:
@ DEVICE PIC12F683, FCMEN_OFF, IESO_OFF, INTRC_OSC_NOCLKOUT , WDT_OFF, PWRT_ON, MCLR_OFF, PROTECT_ON, BOD_OFF
 
 OSCCON   = %01110001         ' Internal 8MHz osc.
 DEFINE OSC 8
 
 ' ======= Common Settings =================================
 ADCON0 = 0
 ANSEL   = 0
 CMCON0 = 7                     ' Comparators off.
 
 TRISIO = 0
 GPIO     = 0
 
 ' ============= PWM Settings =========
 CCP1CON =     %00001100             ' CCP1, PWM mode
 PR2         =     250                         '
 T2CON     =     %00000101             ' TMR2 on, prescaler 1:4
 CCPR1L   = 0
 Speed VAR CCPR1L ' Speed variablefor LED brightness.   
 
 Speed2 VAR BYTE   ' Speed2  variable for fan speed.
 
 ' GPIO.0         ' Empty.
 ' GPIO.1         ' Empty.
 ' GPIO.2         ' LED is connected to GPIO.2 (on board PWM module)
 Btn   VAR GPIO.3   ' Assuming ; the pin is pulled down to GND by 10K resistor, and button pulls it high to Vdd.
 Btn2 VAR GPIO.4   ' Second button. But the pin is pulled high internally. So no external pull up or pull down resistor is required.
                  ' Have a button between this pin and GND (unlike the first button).   
 Fan   VAR GPIO.5   ' Fan motor output. 
 
 
 INPUT Btn2             ' Make the second button pin an input pin.
 OPTION_REG.7 = 0 '
 WPU.4 = 1               ' Enable internal pull up resisitor for Second button pin (GP4)
 
 
 
 PAUSE 10
 Speed2 = 0
 
 ' First button, btn, controls the LED via onboard PWM module.
 ' Second button controls the FAN via software PWM via GPIO.5 output pin and GPIO.4 input pin.
 ' Having the Fan on GPIO.2 and LED on GPIO.5 would be better I think.
 
 Start:
 
          IF Btn = 1 THEN           ' When this button  is pressed, it pulls the pin high.
                 IF Speed + 85 < 256 THEN       ' 0, 85, 170, 255, 0 
                          Speed = Speed + 85
                 ELSE
                         Speed = 0
                 ENDIF
         
         WHILE Btn = 1           ' When this button is being pressed, have no interruption on Fan control.
                         PWM Fan,Speed2,1   
                    WEND
     ENDIF
     
     IF Btn2 = 0 THEN           ' When this button  is pressed, it pulls the pin low.
                 IF Speed2 + 85 < 256 THEN       ' 0, 85, 170, 255, 0 
                          Speed2 = Speed2 + 85
                 ELSE
                         Speed2 = 0
                 ENDIF
         
         WHILE Btn2 = 0                 ' When this button is being pressed, have no interruption on Fan control.
                         PWM Fan,Speed2,1    
                     WEND
     ENDIF
     
     
     PWM Fan,Speed2,1   ' Always drive the fan output.  
     
     
 GOTO Start
 
 END