The two specs explained
Case diameter is the simplest spec: the width of the case measured horizontally, excluding the crown. A 40mm watch has a 40mm-wide case. Lug-to-lug is the vertical measurement from the tip of the top lug to the tip of the bottom lug; it determines how far the watch extends across your wrist's top-to-bottom width. Most watches have a lug-to-lug 5-12mm larger than their diameter.
Why lug-to-lug matters more
Your wrist is wider (circumference) than it is tall (top-to-bottom width). A watch that overhangs the top or bottom of your wrist looks oversized regardless of its diameter. The Tudor Black Bay 58 wears smaller than its 39mm diameter suggests because its lug-to-lug is only 47mm. The original Black Bay 41 wears bigger than 41mm because its lug-to-lug is 50mm+. When trying watches, slip them on and look at how the lugs sit relative to your wrist edges.
When diameter wins
For dial visibility and visual presence, diameter is what you see across a room. A 36mm Datejust looks 'small'; a 44mm Pelagos looks 'big'. The dial real-estate is governed by diameter, not lug-to-lug. So if you want a watch that visually dominates the wrist, prioritise diameter; if you want one that fits comfortably without rocking, prioritise lug-to-lug.
The thickness factor
A third often-ignored spec is thickness: anything over 13mm catches cuffs and reads as 'thick'; under 11mm reads as elegant. Modern integrated sport watches often run thick (the Royal Oak at ~10mm is unusually thin for the genre); modern dive watches commonly run 14mm+. Try the watch under a shirt cuff before buying if you wear long sleeves.