Modifications to a game alter the way it behaves, but more specifically to the context of this question, it is doing so at the asset level.
You alter the files to get the expected modification.
If the new version of the original game alters any of the assets we are modifying, the modification will not behave as it did originally, and that is why out of date mods tend to have issues.
I can personally vouch that this does not always make a difference, however, my experience comes from other games.
Martin Sojka insists that 1.6.4 and 1.7.2 vary so greatly from each other that, if not for the nature of Minecraft, they would be given different titles.
If this is the case, minimalistic mods may still work, i.e. the ones that did not really do much to begin with. If they automated a few command lines, or changed an interface that is still there, that will likely still all work as intended. But any mod worth the time will likely cause issues, as it will effectively be talking a different dialect of the same language to Minecraft.
To answer your second question, the only way to get around any issues that come out would be to modify the modification to support the newer version. This is most likely not something you should do without a great amount of knowledge on what your doing.
My recommendation past that is to check the official site for the mod developers. Find out if they plan on releasing a newer version, and if not, maybe contact them and ask for an update. At the very least, they would have better knowledge of their own product to give you advice on using it in a newer version.