How would an ensnaring strike interact with attacking with a net?

would the target have to make a strenth check and a strength saving throw at the same time on their round and if so would these interact with each other in any way? 

I've put the descriptions below for convenience.



Net-
A Large or smaller creature hit by a net is Restrained until it is freed. A net has no Effect on creatures that are formless or creatures that are Huge or larger. A creature can use its action to make a DC 10 Strength check, freeing itself or another creature within its reach on a success. Dealing 5 slashing damage to the net (AC 10) also frees the creature without harming it, ending the effect and destroying the net. When you use an action, Bonus Action, or Reaction to Attack with a net, you can make only one Attack regardless of the number of attacks you can normally make.

Ensnaring strike-
The next time you hit a creature with a weapon attack before this spell ends, a writhing mass of thorny vines appears at the point of impact, and the target must succeed on a Strength saving throw or be restrained by the magical vines until the spell ends. A Large or larger creature has advantage on this saving throw. If the target succeeds on the save, the vines shrivel away.

While restrained by this spell, the target takes 1d6 piercing damage at the start of each of its turns. A creature restrained by the vines or one that can touch the creature can use its action to make a Strength check against your spell save DC. On a success, the target is freed. 

I see no reason a creature can’t be restrained by two separate sources simultaneously. Nets are hard to hit with, given that they have short range and always attack at disadvantage, but if you’re comfortable that you can land a blow, seems like a decent way to ensure they get restrained and stay that way for more than one round.

I'm using it with sharpshooter at long range (10/15ft) to get around the net having disadvantage. What I'm thinking is would being restrained twice impose disadvantage on either throw or would there have to be an order to make the throws because ES says the creature makes a saving throw when the attack happens and the net is a strenth check on the targets turn. I'd expect that the net and the ES entangle and layer up.

You can certainly be restrained by more than one thing at a time (unless they are both from the same feature because things of the same name don't stack) - though in that case you won't have two instances of Restrained on you. It will be one instance - but you will have to break free of both effects that are making you Restrained before you lose it.