Many spider-like creatures (cf. giant spider) have the ability web walker , which says:
Web Walker. The [creature] ignores movement restrictions caused by webbing.
What sorts of features and conditions does this ability ignore?
Both the web spell and web-filled areas as a dungeon hazard (DMG 105) are treated as difficult terrain, and that seems an intuitive place to start. But the definition of difficult terrain tells us:
You move at half speed in difficult terrain--moving 1 foot in difficult terrain costs 2 feet of speed--so you can cover only half the normal distance in a minute, an hour, or a day.
To be pedantically precise, difficult terrain does not restrict movement per se, it reduces speed (and increases the speed cost of movement). To be more charitable, even though it does not directly restrict movement, it indirectly does so by restricting the total amount one could move in a turn to half of what one normally would. Thus we can assume that a creature with web walker would ignore the difficult terrain effects of webs.
However, both the web spell and the Webs hazard also force a save each turn to avoid the Restrained Condition:
A restrained creature's speed becomes 0, and it can't benefit from any bonus to its speed.
Attack rolls against the creature have advantage, and the creature's attack rolls have disadvantage.
The creature has disadvantage on Dexterity saving throws.
While the first effect of the restrained condition does not reference a "movement restriction", it is a speed reduction in the same sense that difficult terrain is.
Suppose a creature with web walker entered an area of webbing (whether from the spell or hazard should not matter). As soon as it entered, it would be forced to save. Regardless of the result of the save, it would not be affected by the difficult terrain. If it made the save, it would not be restrained.
But if it failed the save, am I correct that it would not be affected by the first consequence of Restrained (speed = 0), only by the second and third?
If it subsequently spent an action to break free (of the spell) or escape (the hazard), that would remove the second and third effects (if successful), but regardless of the outcome of the check it would have no effect on the fact that it could ignore the first effect?