"Both are exactly the same image but I get a difference."
But they're not.
You can see this, using the code below for example:
def show_diff(img1, img2):
diff = Image.new("RGB", img1.size, (255,255,255))
for x1 in range(img1.size[0]):
for y1 in range(img1.size[1]):
x2 = img1.size[0] - 1 - x1
y2 = img1.size[1] - 1 - y1
if img1.getpixel((x1,y1)) != img2.getpixel((x2,y2)):
print(x1,y1,x2,y2)
diff.putpixel((x1,y1), (255,0,0))
diff.show()
img_r = Image.open("img/pacman-r.png")
img_l = Image.open("img/pacman-l.png")
show_diff(img_r, img_l)
Which results in

(Here, any pixel that differs between the two images is colored red.)
Or with
def show_delta(img1, img2):
diff = Image.new("RGB", img1.size, (255,255,255))
for x1 in range(img1.size[0]):
for y1 in range(img1.size[1]):
x2 = img1.size[0] - 1 - x1
y2 = img1.size[1] - 1 - y1
p1 = img1.getpixel((x1,y1))
p2 = img2.getpixel((x2,y2))
p3 = round((p1[0] / 2) - (p2[0] / 2)) + 128
diff.putpixel((x1,y1), (p3,p3,p3))
diff.show()
img_r = Image.open("img/pacman-r.png")
img_l = Image.open("img/pacman-l.png")
show_delta(img_r, img_l)
which results in

(Here, equivalent pixels are gray while a white pixel signifies a pixel in img1 was set (dark) while unset in img2 and a black pixel signifies the opposite.)
It seems like you suspected that PIL's Image.transpose method caused the problem, but the source images aren't just transposed.
Image.transpose works as you'd expect -- so something like:
def diff(img1, img2):
im1 = img1.load()
im2 = img2.load()
images_match = True
for i in range(0, img1.size[0]):
for j in range(0, img1.size[1]):
if(im1[i,j] != im2[i,j]):
images_match = False
return images_match
img_r = Image.open("img/pacman-r.png")
# NOTE: **NOT** Using img_l here
print(diff(img_r, img_r.transpose(Image.FLIP_LEFT_RIGHT).transpose(Image.FLIP_LEFT_RIGHT)))
returns True.
(Here, an image is compared to a twice-transposed version of itself)