Considering checking out some of the Savage World supplements - Savage World Horror Companion and Realms of Cthulhu. Both have discussions on how you can have Powers, but at the same time still maintain an atmosphere of horror, and provide some Setting Rules you can consider. You may also want to check out Rippers, which is about a secret order dedicated to fighting the supernatural set in Victorian London.
Restricting Powers
Powers in Savage Worlds tend to be quite powerful. You can restrict Powers by making Power Points valuable - Go through the available Edges with a fine comb, and remove those that make the Power Points economy moot, such as the Wizard professional edge and any edges that give quick regeneration. You can also limit the number of times the players can boost their total Power Points.
Also consider removing Powers that may not fit with your theme - Invisibility and Fly are two examples. You may also want to handicap Bolt and Blast -- Realms of Cthulhu imposes a flat penalties to those spells, for instance.
Making Powers Hard and Dangerous to get
Realms of Cthulhu require the investigators to get their hands on magic by acquiring and reading tomes -- a feat by itself, providing that reading the tomes of powers doesn't drive you insane. Worse, it also costs some of their sanity.
For other power types, consider similar drawbacks. A psion may acquire mental stress as he hones his skills or making himself more vunerable to demonic possession or the like.
Using Powers can Be Dangerous
The core Savage World rules have certain variants on Powers causing backlash on a fumble. You may want to use rules for hitting a TN to cast a spell instead of spending power points, as this make magic more mysterious, and that higher end effects are harder to get.
Escalate your Villains
Another idea to consider is that while the player-characters may have Powers, that doesn't really even the board for them. Consider that in the TV series Supernatural, there is a food chain of sorts among monsters. Leviathans > Angels > Demons > Monsters > Common Human. Your common man may elevate himself to the level of monsters through Powers, but there are bigger predators out there.
Ensure that combat is not the solution
Also think of what's really terrifying, cases where even having the firepower of a battleship is useless. Can Bolt stop the bad guys from poisoning a town water supply which would turn the townsfolk into mindless zombies? What's good is Blast or Bolt if you can't find the creature to begin with and he moves in perfect silence? Some of the best horror movies have cases where violence isn't the answer. I'll recommend the Supernatural series mentioned, and if you like Japanese anime, the Mononoke series (NOT the movie).
Essentially, require the PCs to defeat the bad guys through investigation, research and putting clues together. Mononoke is outstanding in that aspect is that each demon/ghost can only be slain by the hero's 'demon-killing sword', only if he knows what's the form of the demon, what it is after and how it came to be.
Update: Regarding monsters vulnerability
Mix it up with monster vulnerabilities. In Supernatural, there are a few times when
monster vulnerabilities didn't work out - the writers has to make sure there's
a show worth watching! For example, the Colt didn't kill Lucifer, holy fire only sends
an angel away for a while, Ruby's demon killing knife didn't kill Alastair, Alpha
werewolves are immune to silver bullet, smart demons notice Devil's Trap in times
and there's no way in Hell you are going to kill Death.
There is also the hard part of actually striking the monster with their vulnerability. Demons are vulnerable to cold iron? But when they have Quickness and Level Headed, do you think you can get to him before he can get to you?
Another thing to consider: never, never let your supernatural horrors fight your heroes fair. That's what horror is, isn't it? Unexpected terror. They pounce at you when it is least expected, stalk you in the dark, and fight dirty.
Emphasis the difference
If you want to emulate the "oh no we are like insects unto them", whenever a player gets into a physical conflict (trying to hit with a gun, smite with a spell, but not when trying to bluff or outsmart them), all supernatural goons are considered as Wild Cards...and the players aren't...
Best Answer
Sounds to me like the Whispering Vault. That, too, was a horror game that came out in the early '90s — Wikipedia says the first edition came out in 1993, and that jibes with my memory. The ability to choose your physical form came up in a prominent way, as the PCs spend a lot of time in the realm of Dreams, which matches the description of "another universe or plane." The general mission was to eliminate threats from these other realms, the Unbidden.