Many of the uses of a True RNG fall into the general category of generation, without persistent storage, of a value that is different with high probability from any value determined otherwise.
A value that is different with high probability from any value determined otherwise is very useful in cryptographic protocols. For example, under classic CBC encryption with multiple messages enciphered with the same key, an IV needs to be distinct from a previous IV (which is necessary to conceal a possible repeat of the plaintext), and distinct from the XOR of the first block of plaintext with any value that has or will ever enter the input of the block cipher (which is necessary to ensure confidentiality of that first block of plaintext under the assumption that all other plaintext is known).
"Without persistent storage" requirement rules out a Pseudo RNG, and greatly simplify things: in the case of PRNG, persistent storage needs to be made confidential and/or integrity-protected, which is plain impossible on a regular PC under the basic "maid boots USB stick" security threat. Sometime there is just no persistent storage (boot from CD-ROM), or it is a bit slow.
Another reason to use a True RNG is protection of the implementation of a cryptographic algorithm from side-channel attacks, a process often called "masking". For example, protection against DPA of the crypto-engines used in Smart Cards uses random data for that purpose. Using a Pseudo RNG here would create a chicken-and-egg problem (since secure PRNGs use cryptographic algorithms); while this might be solvable, it is simply easier and much faster to use a TRNG.