Is there any reason to change the ISO manually, rather than have it set automatically?

The primary reason to set ISO, along with shutter time and aperture, manually would be to totally control exposure manually rather than let the camera set exposure. Not every scene needs to be rendered with an overall average brightness of medium gray. Left to its own devices, that is what the camera will attempt to do with every frame. Changing exposure compensation or metering mode can sometimes do the same thing, but it's often easier to get consistent exposure from frame to frame under difficult lighting conditions by using fully manual exposure mode to keep Tv (time value or exposure time a/k/a shutter speed), Av (aperture value a/k/a f-number), and ISO the same for every frame.

Beyond that, there can be plenty of other reasons:

• You are using manual flash. With any "auto" exposure mode the camera has no idea how much light you are adding and will attempt to expose the scene "properly" using ambient light.
• You are using TTL flash and would prefer use more flash power to light your subject brighter than the ambient light rather than use higher ISO to boost the ambient light and use the flash only as a "fill" light.
• You actually want a longer exposure time than the camera's automated routines will give you. There are plenty of types of long exposure photography where this would be the case.
• Then there is the case when you are photographing a static scene with a tripod mounted camera and you don't want the camera to shorten the exposure based on the focal length lens you are using. Many cameras' automated exposure routines will base the maximum allowable exposure time on the focal length of the lens used because it assumes the camera is being handheld. Even if you have set the Tv manually, some cameras will use "safety shift" to avoid potential camera shake.
• You want to batch process a large number of images and use the same raw conversion settings, particularly noise reduction settings, to all of them.
• You use a camera that doesn't treat "partial stop" ISO settings as an actual different amplification level, but instead use a "push/pull" or "pull/push" method to set amplification to the nearest full stop ISO setting. In such a case you might prefer to have full stop ISO settings (100, 200, 400, 800, 1600, etc.) and -1/3 stop ISO settings (160, 320, 640, 1250, 2500, etc) available but also want to avoid +1/3 stop ISO settings (125, 250, 500, 1000, 2000, etc.). For how this works out practically, please see: Is it really better to shoot at full-stop ISOs?

ISO: higher ISO captures more light (as long as it's not clipping/peaking).

The only two variables controlled by the camera that determine how much light is captured are exposure duration (Tv) and aperture (Av). ISO settings affect how much the analog signals collected by the sensor are amplified, not how much signal is collected by the sensor.

Please see Why would using higher ISO and faster shutter speed yield more noise than using lower ISO and slower shutter speed?

In that question, two sets of exposure settings are compared:

• ISO 1600 at 1/125 second with a constant aperture setting.
• ISO 3200 at 1/250 second with a constant aperture setting.

In the second case half as much light was collected by the camera's sensor, then amplified by twice as much before analog to digital conversion. Increasing amplification of the signal also increases amplification of any noise in the analog signal.