Regarding the primary meaning, there's probably no semantic difference worth noting, so OP's example is effectively tautological repetition for stylistic purposes. In terms of actual usage, note that "to rob" is becoming increasingly less common - so if you're unsure which to use, go for "to steal" by default. The main syntactic difference is probably best illustrated by...
1: I robbed my brother's wife
2: I stole my brother's wife
...where the meaning of #1 is I [illegally, secretly] took something from my brother's wife, whereas #2 means I [illegally, secretly] took my brothers wife - probably, from my brother. That's to say, the "direct object" of to rob is usually the owner of whatever you took illicitly, whereas the direct object of to steal is always the thing illicitly taken.
The above distinction has clear implications for passive constructions...
3: I was robbed
4: I was stolen
...where #3 means someone illegally took something from me (or colloquially and slightly more figuratively, circumstances prevented me from getting what I should have been entitled to). And the somewhat unlikely #4 would mean I personally was illicitly acquired (I was treated like "property").
EDIT: There's a good point in @kasperd's comment above - rob more strongly evokes the existence of a victim. Often, the victim of a robbery is forcibly divested of possessions - usually they're at least aware they've been robbed. But stealing is often far less noticeable to the victim. Thus, given...
5: Tom robbed the bank yesterday
6: Dick stole from the bank yesterday
...we're naturally inclined to imagine Tom held the bank up at gunpoint, and is now either in prison or on the run. But we probably assume Dick committed some kind of "white-collar crime" (maybe a computer-based book-keeping fraud) that might not yet (or ever) be noticed by the bank.
TL;DR: If someone robs you, you usually know it, but if they steal from you, you might not.
I don't know that there's a meaning difference between them in that use.
What there is, is a sort of "formality" difference. "Allow" sounds more formal to me than "let". Using "let" is relaxed while "allow" sounds a bit stiff.
One note, though... using allow in the sentence as-is is actually incorrect. You need to add "to":
Jim's parents wouldn't allow him to go to the demonstration.
Conversely, as Colleen rightly points out, "let him to go" would be incorrect. So, your options are:
Jim's parents wouldn't allow him to go to the demonstration.
Jim's parents wouldn't let him go to the demonstration.
Best Answer
If I am pardoned, I am no longer subject to the consequences of having been found guilty (e.g. of a crime). If I am forgiven, I am no longer the cause of negative emotions in the one who forgave me. You can say that pardoning is generally related to a change in punishment or consequences, while forgiving is generally related to a change in feelings or perspective. There is some overlap between the two words, though.