You are misunderstanding a vital element of the rules. There are not two different kind of saves.

What you're reading as "Spell Save" is actually "Spell Save DC". It is used to calculate the save difficulty of spells casts by your character. Spell save difficulties are not set by the DM.

When you cast a spell that calls for the target to make a saving throw, the DC is equal to 8 + your proficiency modifier + the appropriate ability modifier. The target of the spell then makes a saving throw against that DC.

For example, a level one Wizard with 16 intelligence casts Burning Hands at a target. This would have a spell save DC of 8 + 2 (proficiency) + 3 (ability modifier) = 13. Therefore, the target must make a dexterity saving throw against a DC of 13 or take the full effects of the spell.

All saving throws are made by rolling d20 + the appropriate ability modifier, versus a DC. If the target has proficiency in the appropriate save, they also add their proficiency bonus. So, in this case, if the target were a level one Rogue with 16 dexterity and proficiency in dexterity saves, they would roll d20 + 2 (proficiency) + 3 (ability modifier). If the results of that roll equal or exceed the DC of 13, the save succeeds.

Persistent spell simply allows a spell to invoke two saving throws instead of one. If the target fails either of the saving throws, they suffer the full effects of the spell unless they have the ability Evasion or Improved Evasion (in the instance of spells requiring Reflex saves)

In the instance of Constricting coils, a Will save is triggered and if failed the target is subject to Paralyzation and constriction. Since the spell is persistent, whenever a will save is attempted (at the beginning of that round) two saves must be rolled instead to determine if the target can break free of the paralysis/constriction.

In the instance of Black Tentacles, that spell does not have a saving throw, therefore it is not subject to being Persistent. At the bottom of the Persistent spell metamagic feat, it reads:

Spells that do not require a saving throw to resist or lessen the spell’s effect do not benefit from this feat.