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Is there any difference between "忍受不住" and "忍受不了" in terms of their usages?
for example,

忍受不了噪音

忍受不住噪音

or

我再也忍受不了隔壁的噪音了。

我再也忍受不住隔壁的噪音了。

忍受不住: you can't hold it up.

忍受不了: you can't stand it or I've had had it.

忍受不住 takes the abstract things(such as feelings) as its object. E.g. 忍受不住诱惑; 忍受不住疼痛. For the normal object like 噪音, you should use 忍受不了 instead. E.g. 我受不了你了！; 我受不了这噪音了！

So, the correct sentences from your examples should be:

忍受不了噪音

我再也忍受不了隔壁的噪音了。

忍受 = to endure/ to stand (put up with)

Your question is about the difference between two particles '住' and '了'; It also involve the usage of a potential particles '得' (see How to determine the part of speech tags of words succeeding 不 like these?)

住 as a particles denotes a verb is in a continuous state:

忍受住 = to endure (continuously)

忍受得住 = able to endure (continuously)

忍受不住 = unable to endure (continuously)

~

了 as a particles denotes a verb is completed

忍受了 = have endured

忍受得了 = able to endure (successfully)

忍受不了 = unable to endure (successfully)

bkrs：不住 1) unable to v. (also repeatedly
continuously
constantly)，unable to (resist, conceal etc)
疼极了！我忍不住了！ It's awfully painful! I can't stand it anymore 忍不住
special cases having their own dictionary entries 禁不住（承受不住） be unable to bear or endure:
这种植物禁不住冻。 This plant can't stand frost.
你怎么这样禁不住批评? How is it that you can't stand a little bit of criticism?
薄冰禁不住卡车重量。 The thin ice cannot bear truckloads.
（抑制不住; 不由得） can't help (doing sth.); can't refrain from:
禁不住笑了起来 can't help laughing; burst out laughing
按捺不住to be unable to hold back
àn nà bú zhù
beside oneself (with excitement (anger; joy)); cannot control (contain) oneself; unable to contain any longer; cannot contain (control) one's excited feelings; cannot hold back; unable to hold oneself in check:
按捺不住激动的心情 be unable to hold back one's excitement
她此时按捺不住自己，料想一定是凶多吉少。 She could now no longer contain herself， and was convinced that some disaster was impending.
听到那个好消息时，他按捺不住高兴的心情。 When he learned of the good news, he was beside himself with joy. 经不住 see 禁不住 similarly users may want to look up the following special compounds
靠不住 对不住 保不住 架不住 闲不住 挂不住 招架不住 吃不住 备不住 搁不住 打不住

不了 unable to
（also without end) find many more special cases of verb + 不了 at bkrs, however users note that 忍不住 has an entry in bkrs (as one word) with frequency rank #2326 ! whereas 忍不了 does not have indicating that the former must be more common,

regarding 忍受不了 忍受不住，bkrs 忍受不住 be unable to stand (sth.) apparently still more frequent than 忍受不了 which does not have an entry as a single word

To me, both have the very similar meaning. If there were any difference, it would be too trivial.