The larger image contains a 15204-byte iTXt chunk containing some Adobe metadata. The smaller image also contains a similar chunk but it's smaller, only 866 bytes. As Mario's answer says, you can use pngcrush to remove it.
Get "pngcheck" and run "pngcheck -v file.png" to find out what's in the file.
The 16.4KB file:
$ pngcheck -v 6*
File: 6iutM.png (16643 bytes)
chunk IHDR at offset 0x0000c, length 13
32 x 64 image, 8-bit palette, non-interlaced
chunk pHYs at offset 0x00025, length 9: 2835x2835 pixels/meter (72 dpi)
chunk iTXt at offset 0x0003a, length 15204, keyword: XML:com.adobe.xmp
uncompressed, no language tag
no translated keyword, 15183 bytes of UTF-8 text
chunk cHRM at offset 0x03baa, length 32
White x = 0.31269 y = 0.32899, Red x = 0.63999 y = 0.33001
Green x = 0.3 y = 0.6, Blue x = 0.15 y = 0.05999
chunk PLTE at offset 0x03bd6, length 768: 256 palette entries
chunk tRNS at offset 0x03ee2, length 7: 7 transparency entries
chunk IDAT at offset 0x03ef5, length 506
zlib: deflated, 32K window, maximum compression
chunk IEND at offset 0x040fb, length 0
No errors detected in 6iutM.png (8 chunks, -712.6% compression).
The 1.52KB file:
$ pngcheck -v 7*
File: 7W95m.png (1523 bytes)
chunk IHDR at offset 0x0000c, length 13
32 x 64 image, 8-bit palette, non-interlaced
chunk tEXt at offset 0x00025, length 25, keyword: Software
chunk iTXt at offset 0x0004a, length 886, keyword: XML:com.adobe.xmp
uncompressed, no language tag
no translated keyword, 865 bytes of UTF-8 text
chunk PLTE at offset 0x003cc, length 21: 7 palette entries
chunk IDAT at offset 0x003ed, length 498
zlib: deflated, 32K window, maximum compression
chunk IEND at offset 0x005eb, length 0
No errors detected in 7W95m.png (6 chunks, 25.6% compression).
1This is very interesting. I'd like to know how this could have happened. It definitely isn't related to the contents or resolution or anything like that, this is caused by the software handling the compression (e.g. photoshop). the question is whether it's a bug or if it's intentional. – Cestarian – 2016-03-08T18:56:05.967
For the person who edited my post, thank you. But you are wrong on kB, it's with a capital K since it's not your standard 1000's prefix it stands for 1024 and the uppercase was introduced a long time ago when bits where introduced. – Madmenyo – 2016-03-08T22:32:13.340