Hi Osee,
As applied to problems, some would consider them the same, to the extent that problems should be made to go away. "Dispose of" represents the more negative approach, as though you wish the problem had never arisen, and there were nothing to be gained by solving it, other than its disappearance. This reminds one of the "garbage disposal" unit in the sink, or disposing of trash in the trash bin.
"Deal with" treats a problem more like an exercise, or an opportunity to gain something positive. Of course not everyone feels that way. You often hear, "I just can't deal with this anymore!" Or, "Here! It's your problem now. You deal with it."
You may also "deal with" a used car dealer.
We also say, "I'm going to "handle" this problem, which is usually more neutral.
Regards, - A.