Do all opponents have weak points?

UPDATE: The 10/4/2012 patch now allows for 40% bonus to headshots on brutes, banshees, and Geth primes. Also +40% for the rear exhaust port on the Atlas mechs. Text updated below, except that the developer quote is left as originally quoted.

First off, a direct quote from Bioware programmer Brenon Holmes is illuminating, and provides some interesting information on the intended state of vulnerabilities:

Ah... hmm - I'm not sure why your empirical testing would have come up with some of that - but here's how it's "supposed" to be.

Banshee

• No weak points (aim for the belly, as it's fattest - most likely to hit it)

Brute

• Takes reduced damage in the right arm
• Takes greatly reduced damage in the claw
• Armour Plates absorb damage until breaking (best to aim for exposed flesh, probably base of the neck?)
• No other weak points

Geth Prime

• Currently has a head volume (but it's a bug)

Atlas

• Some types of weapons will do increased damage to the cockpit (this is a bug)
• Destroy any of the following to do a small damage spike (shoulder flanges, groin plate, knee plates)

This tells us a few things:

Banshees headshots get a 40% damage bonus. Geth Primes weren't intended to have weak points, but did for awhile. A recent patch enabled intentional 40% damage bonus for headshots. Brutes have 'weak points', but really, they're more like 'less strong points'. But a recent patch enabled a 40% damage bonus for headshots.

In addition, as Rare Candy notes, Turrets, Swarmers, and Drones do not have any weak point.

Aiming for the head is a pretty good idea. There are a few notable exceptions and additions, even here however. Most of these notes are gleaned from the Prima guide, or from experience. I've verified most of these myself in game, and done a fair amount of testing on most of them, so I'm fairly confident here.

Cerberus Troops

• Combat Engineers carry their turrets on their backs. If you can catch them from behind before they deploy the turret, and hit it hard enough, it's possible to cause it to explode for a one-shot kill.
• Phantoms swords can be destroyed by a well placed shot, eliminating both their ability to inflict one-shot kills on your squad, and their ability to deflect incoming fire, making them a bit easier to take down.
• As noted above, Atlases are vulnerable at a number of key points, including the shoulders, knees, groin, and rear exhaust port and thrusters on the back. Additionally, the Cockpit is vulnerable, but this is a bug. _{Devs have indicated that it's likely a bug that they will probably 'keep', but at present, it works rather inconsistently, so expect it to be patched in some way.}

Reapers

• Ravagers don't quite have a 'head' per se. Instead, they are vulnerable to direct hits to the blue glowing circle in between their two front-cannons. Landing a shot right in the bulls-eye can have a devastating effect. Be careful about puncturing the sacs at close range, as that will release Swarmers.
• Harvesters (SP only) also have a similar glowing blue target, right on the front of their belly. I find lodging a few sticky grenades from a Scorpion onto this spot to be 1) very easy, and 2) very effective. YMMV however.
• Brutes have visible armor plating which provides substantial damage reduction. This can be overcome through the use of Shredder Mods and Armor Piercing Ammo, in much the same way as a Guardians Shield, or by carefully aiming around it, to hit the exposed articulated head or neck, or by breaking the armor plates with repeated fire (inadvisable).

Geth

• Geth Rocket Troopers, if you're incredibly good (Or playing an infiltrator or Soldier on SP), are succeptible to having their rockets shot out of mid-air. If you hit them fast enough after launch, the explosion can kill the Rocket Trooper, and any friends he might have nearby. This is... not easy to do intentionally however.
• Geth Pyros carry a large visible fuel tank on their back. This fuel tank is highly explosive. You know what to do.

Turrets, swarmers, and combat drones are the only things in the game that don't have weak points.

Shooting Geth in the head deals more damage and can earn you headshots in multiplayer. Do headshots do extra damage to synthetic enemies?

Likewise with shooting brutes, banshees, marauders, and cannibals. (Brutes it can help to shoot off their shoulder heads)

Ravagers are vulnerable to being shot in the sacks.

Cerebus Atlas has a weakpoint in the cockpit, like the codex says.