For example, with Dual EC DRBG, researchers discovered that it would be possible to create the constants based on another constant, with which you could predict the output easily. But without prior knowledge of that constant, it would be an infeasible brute-force search to find it.
Likewise, previous publicly known backdoors like the one in the export version of Lotus Notes depended on a key that the NSA had. There it was even simpler, and not obfuscated; it would just encrypt a portion of the session key with the NSA's public key, which they could decrypt and the easily brute-force the rest of the session key.[1]
The NSA doesn't want to make security weak against arbitrary attackers, they just want to give themselves the keys.
[1]: http://www.cypherspace.org/adam/hacks/lotus-nsa-key.html
Another point is that most people (especially non-US citizens) don't necessarily view NSA and NIST as separate. They are seen as part of the same government. More like 2 offices in the same government department.
Now this also brings about an interesting thing I have been thinking about. NSA is also in charge of protecting its own data. So recommendations, practices and policies they tweak go into keeping its (and other agencies') classified data secure.
Given that they have managed to "tweak" and insert backdoors in some algorithms or systems, how likely they are to recommend those systems for its own and other government agency use? Do they want the communication or keys to the nuclear launch sites to use the "tweaked" version. They would need to have an pretty good feel that no other agency out there has also figured out the back door.