Build new dataframe with for loop given condition in another dataframe

You can achieve this quite easily via grouping functions.
Unfortunately the first and last values of the three countries in your dataset are the same, so the result is also three times the same two values.
(Perhaps there is sth wrong with the sample data?)

First group the dataframe by countries and pick just the two columns of interest:

grpd = df.groupby('countries')['gini', 'ECONOMIC FREEDOM']

With this Groupby-Object you can apply functions to the the subsets of your data which are separated by the grouping feature, countries in your case.
E.g. to get the last value in each group just ask for

grpd.last()

           gini  ECONOMIC FREEDOM
countries                        
Argentina  50.7              6.16
Bolivia    50.7              6.16
Chile      50.7              6.16

or accordingly for the first row per group

grpd.first()

           gini  ECONOMIC FREEDOM
countries                        
Argentina  40.8              4.25
Bolivia    40.8              4.25
Chile      40.8              4.25

for calculating the percentage of change of the last with respect to the first you could therefore simply write

(grpd.last() - grpd.first()) / grpd.first()

                gini  ECONOMIC FREEDOM
countries                             
Argentina  0.242647         0.449411
Bolivia    0.242647         0.449411
Chile      0.242647         0.449411

EDIT:
the output can also be formatted, e.g. like:

df_change = (grpd.last() - grpd.first()) / grpd.first()

df_change.applymap(lambda x: str.format('{:+.1f%}', x))

             gini ECONOMIC FREEDOM
countries                         
Argentina  +24.3%           +44.9%
Bolivia    +24.3%           +44.9%
Chile      +24.3%           +44.9%

EDIT2:
for signs only:

df_change.applymap(lambda x: ['-', ' ', '+'][np.sign(x).astype(int)+1])

          gini ECONOMIC FREEDOM
countries                      
Argentina    +                +
Bolivia      +                +
Chile        +                +

Create empty lists and append desired values from countries, gini, and ECONOMIC FREEDOM columns for each country.

    countries = []
    gini = []
    efw = []
    for i in new_df.index.levels[0]:
        countries.append(i)
        country = new_df.loc[i]
        country = country.reset_index()
        x = country.iloc[0].tolist()
        y = country.iloc[-1].tolist()
        change_g = (((y[2] / x[2]) - 1) * 100)
        change_e = (((y[3] / x[3]) - 1) * 100)
        gini.append(change_g)
        efw.append(change_e)

Then do a for loop. For each number you append a + or a -.

g = []
e = []
for n in gini:
    if n > 0:
        g.append("+")
    g.append("-")

for f in efw:
    if f > 0:
        e.append("+")
    e.append("-")

Then create a dataframe with the lists countries, g, and e.

tuples = list(zip(countries,g,e))
changes = pd.DataFrame(tuples, columns=['Country','Change in Gini', "Change in Economic Freedom"])