If your A/C is turned off then it is protected against LOW voltage. High voltage it is most likely safe from less it is high enough (not likely).
The 10 dollar device you mention is a spike suppressor, all surge protectors contain them so do many other devices, A SPIKE is a very short duration voltage spike (Can't really call it a surge since it only lasts, usually, 1/120 second or less).
IT is of no use at all in a brownout (low voltage) or true Surge (Sustained high voltage) situtation,, Been hit with one of the latter and those 10 dollar firecrackers sounded like July 4th.
Now, the real danger is this: In the summer if you are using your A/C and you have say 110 volts line with poor wires underground in the park the A/C kicks in and the voltage may drop to 100 or less. This means that the compressor does one of two things, it either takes a long time starting, thus overheating the motor and doing damage, or it stalls (Same effect). Over time this damage builds up and you need to replace teh A/C, about 1,000 bucks.
The RV type Surge guards, the good ones with digital displays, will cut you off, wait 2 minutes and try again.. Thus protecting the compressor against this type of damage.
Next is a true SURGE... This can happen in a park if several A/C's cut out at the same time or if a big ticket electricty user on the same high tension branch shuts off his main power sucker. The voltage now goes not to 120 but perhaps as high as 150 or more and remains there for several cycles.. (A Surge) this can blow electronics, TV, Microwave, Sat Receiver, Converter, and such. OR.. More likely... You plug into a 30 amp outlet THINKING it's for an RV only to find it's a 240 volt hole in the wall... The Surge guard says "Hey Stupid, 240 volts is not good" and blocks power from blowing all your expensive stuff.
Thus paying for itself several times over.