Use a hex editor like @ChrisF suggests. A good one for Windows is "HxD."
Please note that generally computers deal with multiples of 8 bits at a time unless interfacing directly with hardware (and most of the time even then). So looking at a file divided on those boundaries is usually better and more meaningful than just a raw stream of 0s and 1s.
Learn about hexadecimal and how it relates to binary first. It's really pretty simple (0 = 0000, 1 = 0001, 2 = 0010, ... 9 = 1001, A = 1010, B = 1011, C= 1100, D = 1101, E = 1110, F = 1111)
Also, comparing the files by hand is likely to be tedious. You'll probably need a tool like BinDiff, vbindiff, or similar.