Translation:Mom does not allow me to write on the walls.
In English they're both ways of saying the same thing; "will not allow"/"won't allow" can refer to the present tense as well as the future, depending on the context. It's like "My TV won't work" can also mean it's not working in the present.
True. But I don't know if accepting it because of such a nuance wouldn't potentially confuse some people rather than help... this is rather a rarely suggested construction, and I think I'd prefer to reject it. So it is clear that this sentence is in Present Tense.
I wrote "My mom does not allow me to write on walls." It gave two possible solutions: "Mom does not allow me to write on the walls" or "Mom does not permit me to write on the walls." To me, using "the walls" seems to indicate the walls of our house. I took this sentence to mean walls anywhere (at relatives' and friends' homes, etc., even on exterior walls, i.e., no graffiti!). I think this answer should be accepted.