AdaBoost vs Gradient Boosting

These are what I've understood so far:

• There are both boosting algorithms, which learns from previous model's errors and finally make a weighted sum of the models.
• GBM and Adaboost are pretty similar except for their loss functions.

But still it is difficult for me to grab an idea of differences between them. Can someone give me intuitive explanations?

• In Gradient Boosting, ‘shortcomings’ (of existing weak learners) are identified by gradients.
• In Adaboost, ‘shortcomings’ are identified by high-weight data points.