The "Expert" field lists all possible adjustments, and there are lots of them that change the original font:

Font Formats:    TrueType WOFF   WOFF2   EOT Lite  EOT Compressed  SVG

The mere act of changing a font's outlines to another format may change curves and hinting.

Truetype Hinting:    Font Squirrel   Keep Existing   TTFAutohint

Hinting, in a properly made font, is its creator's choice of how the font will display at small pixel sizes. The two options that change it are automated, while the existing hinting may be 'optimal', according to the designer (and decided on a per-character basis).

Rendering:   Fix Vertical Metrics  Fix GASP Table  Remove Kerning

Not sure what may need to be 'fixed' in the vertical metrics. The GASP Table ("Grid-fitting and Scan-conversion Procedure"), again, should be determined by the font's creator. "Remove Kerning" may be a 'fix' for browsers that do not support it, but a good font needs kerning, so removing it indeed decreases the font's overall quality.

Fix Missing Glyphs:  Spaces  Hyphens     

It's up to the font's creator to decide which characters should be in the font. Including 'generic' ones may lead to discrepancies in size (too narrow or too wide spaces) and style (consider a highly stylized font which suddenly contains a very plain hyphen).

X-height Matching: Resize to match the selected font's x-height.

Re-scaling a font is inaccurate, as it's designed on a fixed grid. Therefore, if you scale it, you are bound to loose some accuracy due to rounding.

Subsetting:  Basic Subsetting Custom Subsetting... No Subsetting

In my book, removing characters from a font counts as 'decreasing its quality'.

OpenType Features:  Keep All Features   Comma Feature List 

.. and the same for OpenType features. (I know: most of them are not supported anyway.)

OpenType Flattening: If the features are available, the generator will flatten
them into the font.

Not sure about what 'flattening' means here, but again, it's removing existing features.

Advanced Options: [..] Em Square Value Adjust Glyph Spacing

Again, re-scaling the font to another grid than it was designed on, and changing the overall design.

There can be lots of reasons you may not notice "quality loss", not in the least because you are not looking close enough... The effect of removing 'hinting', for example, can be quite small (when there was not much of it anyway, or applied badly), or extreme, especially in small sizes. The same goes for all operations that scale the original outlines: there are bound to be differences, but they may occur at pixel levels.

Other effects, such as removal of OpenType features and characters, are only noticeable if you happened to expect those features and/or characters.