If a weight for one variable is higher than for another, is it more important?
Yes - if your predictor variables are normalized.
Without normalization, the weight represents the change in the output per unit change in the predictor. If you have a predictor with a huge range and scale that is used to predict an output with a very small range - for example, using each nation’s GDP to predict maternal mortality rates - your coefficient should be very small. That does not necessarily mean that this predictor variable is not important compared to the others.