How do I detect that two images are "the same" even if one has slightly different cropping/ratio?

You may want to take a look at feature matching. The idea is to find features in two images and match them. This method is commonly used to find a template (say a logo) in another image. A feature, in essence, can be described as things that humans would find interesting in an image, such as corners or open spaces. There are many types of feature detection techniques out there however my recommendation is to use a scale-invariant feature transform (SIFT) as a feature detection algorithm. SIFT is invariant to image translation, scaling, rotation, partially invariant to illumination changes, and robust to local geometric distortion. This seems to match your specification where the images can have slightly different ratios.

Given your two provided images, here's an attempt to match the features using the FLANN feature matcher. To determine if the two images are the same, we can base it off some predetermined threshold which tracks the number of matches that pass the ratio test described in Distinctive Image Features from Scale-Invariant Keypoints by David G. Lowe. A simple explanation of the test is that the ratio test checks if matches are ambiguous and should be removed, you can treat it as a outlier removal technique. We can count the number of matches that pass this test to determine if the two images are the same. Here's the feature matching results:

Matches: 42

The dots represent all matches detected while the green lines represent the "good matches" that pass the ratio test. If you don't use the ratio test then all the points will be drawn. In this way, you can use this filter as a threshold to only keep the best matched features.

I implemented it in Python, I'm not very familiar with Rails. Hope this helps, good luck!

Code

import numpy as np
import cv2

# Load images
image1 = cv2.imread('1.jpg', 0)
image2 = cv2.imread('2.jpg', 0)

# Create the sift object
sift = cv2.xfeatures2d.SIFT_create(700)

# Find keypoints and descriptors directly
kp1, des1 = sift.detectAndCompute(image2, None)
kp2, des2 = sift.detectAndCompute(image1, None)

# FLANN parameters
FLANN_INDEX_KDTREE = 1
index_params = dict(algorithm = FLANN_INDEX_KDTREE, trees = 5)
search_params = dict(checks=50)   # or pass empty dictionary
flann = cv2.FlannBasedMatcher(index_params,search_params)
matches = flann.knnMatch(des1,des2,k=2)

# Need to draw only good matches, so create a mask
matchesMask = [[0,0] for i in range(len(matches))]

count = 0
# Ratio test as per Lowe's paper (0.7)
# Modify to change threshold 
for i,(m,n) in enumerate(matches):
    if m.distance < 0.15*n.distance:
        count += 1
        matchesMask[i]=[1,0]

# Draw lines
draw_params = dict(matchColor = (0,255,0),
                   # singlePointColor = (255,0,0),
                   matchesMask = matchesMask,
                   flags = 0)

# Display the matches
result = cv2.drawMatchesKnn(image2,kp1,image1,kp2,matches,None,**draw_params)
print('Matches:', count)
cv2.imshow('result', result)
cv2.waitKey()