Is there any benefit to having all the mods of given level at the same time?

Should I specialize my agents or give them all the same weapons and mods?

1) Nope. Just use the best augmentations available, unless you're really short on money.

2) Specialising is not really necessary or advisable imho. Depends more on how much you like to micromanage, but I don't think the game is meant to be played that way. As all implants are useful to any kind of agent, by not installing some for the sake of diversity you would just gimp your squad.
It may work for the first few missions, but in pratice once you get the Minigun you can easily stick with that for most of the game. Later on you'll face tons of well equipped enemy agents and having a fully upgraded squad with heavy weapons is the only way to go.

One thing I had been considering is upgrading the Brain of only one agent that uses the persuadetron. You only need 1 agent to persuade. The one downside I see is that if you're on an exclusively persuasion mission and that agent dies, mission fail.
Otherwise yeah I typically mod them up pretty similarly and give them the same weapons. Especially because during group selection, clicking a weapon will make other agents who happen to possess that weapon draw it also.

I had that happen to me today. I had persuaded a mob of people - only some of them were civilians, I had lots of cops and enemy cyborgs too - and then I felt cocky with my army behind me, and the one guy with the persuadertron went and died because of my foolishness. All of a sudden *poof* the scanner lights up with a bunch of red dots and gunfire galore.

Needless to say, I failed that mission. But it taught me to have at least two of my squad carry persuadertrons, as a failsafe.