They both deal with computer vision on images. The difference between them is the Custom Vision can only do image classification and object detection, as well as take in your own images. The Computer Vision APIs can do a bit more, but you don't have any control over how the models are trained.
The Computer Vision API is Azure's model and can provide general services for you as below:
Image classification - This is where the API will give you a number of tags that classify the image. It should also give you a confidence score of how strongly the model predicts the image to be of that tag.
Content Moderation - The API can give you an isAdult and isRacy flags to determine if the image meets those criteria. An accompanied confidence score is with those, too.
OCR - The API can read text within the images and will give you the text. This API can also work with handwritten text instead of just text on signs.
Facial Recognition - This API will recognize the faces of celebrities or other well-known people within images.
Landmark Recognition - This will recognize landmarks within images.
Azure Custom Vision is an image recognition service that lets you build, deploy, and improve your own image identifiers. An image identifier applies labels (which represent classes or objects) to images, according to their visual characteristics. Unlike the Computer Vision service, Custom Vision allows you to specify the labels and train custom models to detect them. For object detection, you do the same process, but you pick in the images where the object is you want to detect and give that a tag. Each time you upload and tag new images the model needs to be trained. From there you can evaluate how well your model performs, give it test images, or even use the REST URLs or SDKs to interact with it.
Hope this helps.