Why Is This Day Counter Producing The Wrong Results?

Problem 1:

Your leap year calculation is off:

Leap years are years % 4 == 0 but only for years not year % 100 == 0 unless they are also year % 400 == 0:

2004,2008,2012 : leap year (%4==0, not %100==0)
1700,1800,1900 : no leap year (%4 == 0 , % 100 == 0 but not %400 == 0)
1200,1600,2000 : leap years (* 1200 theor. b/c gregorian cal start)

Problem 2:

In your input you premultiply the year by 365 w/o checking for leap-years - they schould have 366 days but got 365 - which would result in lacking days when computing the amount of days for years that leap(ed).

Problem 3:

You have a controlflow-issue: the get_second_day() repeats because you do:

get_first_date()
    while without end:
        do smth
        call get_second_date(..)
             while without end:
                 do smth 
                 call some calculation functions
                     that calc and print and return with None 
                 back in get_second_date(), no break, so back to the beginning
                 of its while and start over forever - you are TRAPPED

• fix it by putting a break after convert_dates_and_months(date_1, date) inside get_second_day(..)

Suggestions:

You can streamline the input by reducing the amount of duplicate code between get_first_day() and get_second_day() - this follows the DRY principle (Don't Repeat Yourself):

def getDate(text):
    while True:
        try:
            print(text)
            day = int(input('Day:'))
            month = int(input('Month:'))
            year = int(input('Year:'))
            print(day, '/', month, '/', year)
            return [day, month, year * 365]  # see Problem 2
        except ValueError:
            print('You were supposed to enter a date.')


def get_first_day():
    date1 = getDate("First Date")
    # rest of code omitted 

def get_second_day(date_1):
    date = getDate("Second Date")
    # rest of code omitted 

A better solution would utilize datetime and datettime-parsing, especially if you want to handle input validation and leap-year estimation you would need far more checks.

Using datetime module would simplyfy this a lot:

import datetime

def getDate(text):
    while True:
        try:
            print(text)
            day = int(input('Day:'))
            month = int(input('Month:'))
            year = int(input('Year (4 digits):'))
            print(day, '/', month, '/', year)

            # this will throw error on invalid dates: 
            # f.e. 66.22.2871 or even (29.2.1977) and user
            # gets a new chance to input something valid
            return datetime.datetime.strptime("{}.{}.{}".format(year,month,day),"%Y.%m.%d")
        except (ValueError,EOFError):
            print('You were supposed to enter a valid date.')


def get_first_day():
    return getDate("First Date")

def get_second_day():
    return getDate("Second Date")

# while True: # uncomment and indent next lines to loop endlessly
first = get_first_day()     # getDate("First Date") and getDate("Second Date") 
second = get_second_day()   # directly would be fine IMHO, no function needed
print( (second-first).days) 

Output:

First Date
Day:10
Month:9
Year (4 digits):2003
10 / 9 / 2003
Second Date
Day:6
Month:6
Year (4 digits):2006
6 / 6 / 2006
1000 

Good read: How to debug small programs (#1) - following it, could have at least lead you to the control-flow issue.