What does it mean when someone says the NFC tag memory capacity is 1024 byte (NDEF 716 byte)? If I am only using the unique ID of the NFC tag, how do these sizes affect my selection of the tag?
Since you only intend to use the anti-collision identifier (UID), the actual memory size does not affect your application. Most NFC tags have some form of anti-collision identifier. Depending on the tag type, the memory used to store that identifier may already be calculated into the overall memory size or may be stored in a separate memory area that is not counted towards the overall memory size.
The discepancy between the overall memory size and the memory available for storing NDEF data comes from the fact that not all memory regions are usable for storing NDEF data. Depending on the tag platform, some memory areas may be reserved for storing lock bits, capability information, access keys/passwords, the anti-collision identifier, or other meta information. Consequently, these areas cannot be used for general-purpose NDEF data (e.g. to store a website URL).
However, all this does not mean that only using the UID makes you safe to use just any tag.
First, there are tags that are not compatible to all Android devices (in case you intend to use Android as the reader platform, though similar constraints might apply to other reader platforms as well). Particularly the figures you mentioned in your question (1024 byte overall memory, 716 byte NDEF memory) suggest that these are MIFARE Classic 1K tags. These tags use a proprietary protocol that is not available on some Android devices (specifically those without an NFC chipset from NXP). While using the anti-collision identifier would even work on all devices, some manufacturers (e.g. Samsung) decided to explicitly block those tags on many of their devices. Consequently, using MIFARE Classic tags might not be the best choice for your application.
Second, not all tags might expose an anti-collision identifier that's suitable for your application. For instance, there are tags that only expose a random ID that changes with every activation. Also, there might be tags with duplicate IDs. For instance, due to the total amount tags with 4-byte (N)UID that has been manufactured, there must be tags with duplicate IDs.