Right photo - We notice algae, moss, and lichen on the tree branches. Algae and mosses grow in any moist shady site. They live on the tree bark without harming it and no control is needed. Here are photos and more information. https://extension.umd.edu/resource/algae-and-mosses

Lichen are non-destructive living organisms composed of fungus and algae and typically appear as gray-green patches on trees and shrubs. They are harmless and no control is necessary.  https://extension.umd.edu/resource/lichens

Left photo - We notice powdery mildew on the foliage. In general, this is a common late season fungal disease. It is not usually a big problem on Japanese maples. Apparently the weather conditions were perfect this year. 

We do not notice any insects in the photos. There is nothing you can do at this time. When the leaves fall, you can collect them and discard them in your trash to minimize fungal spores overwintering in the area. If mildew starts to become a problem next year, you can decide if you want to manage it with horticultural oil or tolerate it. Powdery mildew can be unsightly but usually doesn't cause lasting or serious damage. Here is more information on the disease, with ideas of how to prevent problems in the future. You may not have an issue next year. https://extension.umd.edu/resource/powdery-mildew-disease-trees-and-shrubs

Pruning - Do not leave a stub as this can cause possible rots, etc. When pruning cut back to a lateral branch or prune back to the branch collar on the trunk. The branch collar is the area of swollen tissue that forms around the base of a branch where it is attached to the tree. See our page on pruning and scroll down to 'Proper Pruning Cuts' for a photo. https://extension.umd.edu/resource/pruning-trees

From what we can see in the photos the Japanese maple looks mature. With Japanese maples most problems are not insect or disease related. It has more to do with growing conditions and location. Branch dieback can be due to several reasons such as a root or trunk problem. 

Check the drainage in the area and make sure the soil drains well. Make sure there are no downspouts dumping water in the root zone. If so, redirect them. Keep mulch no thicker than several inches and away from the base of the trunk. Water during dry periods next season. 

Look for new growth next spring and you can send us photos if you notice additional symptoms. Here is our blog on growing Japanese maples successfully in the landscape. https://marylandgrows.umd.edu/2018/08/24/japanese-maples-in-maryland-landscapes-plant-location-care-are-keys-to-success/

Marian