How does the spectrum of a frequency modulated wave differ from the well known amplitude-modulated one?

FFT of frequency-modulated sinewave

#!/usr/bin/env python
#-*- coding: utf-8 -*-

## Import common moduli
import matplotlib, sys, os, time
import matplotlib.pyplot as plt
import numpy as np
from scipy.constants import c, hbar, pi

## Use LaTeX
matplotlib.rc('text', usetex=True)
matplotlib.rc('font', size=8)
matplotlib.rc('text.latex', preamble = '\usepackage{amsmath}, \usepackage{yfonts}, \usepackage{txfonts}, \usepackage{lmodern},')


## Generating 1D data
x = np.linspace(1., 300*2*np.pi, 10000)
#y = np.sin(x*2*pi) * (1 + np.sin(x*2*pi * .05) * .2)
#y = np.sin(x*2*pi) * (1 + np.exp(1j* x*2*pi * .05) * .2)
y = np.sin(x*2*pi + (1 + np.sin(x*2*pi * .2) * 2.41/2**.5))  * np.exp(-x/300)

## 1D FFT with cropping for useful frequencies
freq    = np.fft.fftfreq(len(x), d=(x[1]-x[0]))         # calculate the frequency axis with proper spacing
yf      = np.fft.fft(y, axis=0) / len(x) * 2*np.pi      # calculate the FFT values
freq    = np.fft.fftshift(freq)                         # ensures the frequency axis is a growing function
yf      = np.fft.fftshift(yf) / np.exp(1j*2*np.pi*freq * x[0])   # dtto, and corrects the phase for the case when x[0] != 0
truncated = np.logical_and(freq>0, freq<2)         # (optional) get the frequency range
(yf, freq) = map(lambda array: array[truncated], (yf, freq))    # (optional) truncate the data points
plt.plot(freq, np.abs(yf), color="#008800", label=u"$y'$", ls='-')                  # (optional) plot amplitude
#plt.plot(freq, np.unwrap(np.angle(yf)), color="#008800", label=u"$y'$", ls='--')    # (optional) plot phase


## Simple axes
#plt.ylim((-0.1,1.1)); 
plt.yscale('log')
#plt.xlim((-0.1,1.1)); plt.xscale('linear')

## ==== Outputting ====
## Finish the plot + save 
plt.xlabel(u"x"); 
plt.ylabel(u"y"); 
plt.grid()
plt.legend(prop={'size':10}, loc='upper right')
plt.savefig("output.png", bbox_inches='tight')