The verb „wiegen“ describes the process of measuring the weight of something, while „abwägen“ means checking a decision, whether this or that comes into question (pro and contra) . If you want to decide whether you want to take a vacation in the mountains or at the sea, then you have to "abwägen" the arguments. If you travel by plane, you should always „wiegen“ your luggage to avoid paying an excess baggage fee.