If I install a ROM that doesn't exactly match my phone, will it work at all? As you know, manufacturers of popular mobile phones often times will release a modified " clone " version of their mainstream models. Often these clone models will be re-released models that are designed to be a downgraded version of a high end phone, usually released for prepaid network. These dumbed down androids will have the exact same hardware as their big brothers, but will be lacking in some performance features such as a smaller lower resolution screen, or a weaker camera. But the are essentially their own kin. They'll have the same chipset, the same sensors and I/O devices, but will be cut down much like an entry level laptop. The problem with these phones is that the manufacturer rarely publishes software updates for them because they aren't very popular and most of the time it's impossible to find third party roms for them for the same reason. So, how likely is it to work if you download and install an OS made not for the prepaid, but for it's big brother?

Say you have a Motorola XT957C3 
This phone is a prepaid, entry level version of the Motorola Droid RAZR MAXX. is uses the Texas Instruments OMAP3040 chipset, just like its cousin, and it has the same Broadcomm WLAN chip, the same Wolfson Audio hub, the same accelerometer and gyro, even the same battery and power IC.
( note that these figures aren't factual, I'm just making an example )
The only differences are purely aesthetic. A smaller display, lack of a front camera, less internal memory etc. etc.

How likely is it that installing a ROM for the Droid RAZR will work on this off-model phone? What other than basic drivers does the OS need to work right? And what about the baseband software? Would it be the same as the one on its brother?
The answer is almost definetly no.

Although there might not be much difference between the models, there are bits that are different. These bits won't work. Other bits may depend on these bits, and they won't work either. The baseband and radios might work fine, but the rest of the mismatched components will lead to, at best, a soft brick. I have a Moto G LTE second gen, and accidentally flashed a ROM from its non-LTE brother. Almost exactly the same apart from the radio, and a different model number of processor (both Snapdragon 400s though). I ended up with a brick that was very hard to sort out (not helped by the fact my nandroid backup was corrupted).

Also (correct me if I'm wrong), I believe that a number of ROMs will refuse to flash if they detect the wrong device. I found that out when an old copy of TWRP kept feeding a zip the wrong information. Don't try and test it out though :)

In short, flashing ROMs on devices they weren't intended to be flashed with is a bad idea, no matter how similar in specs they may seem.