Think of the structure of purines and pyrimidines. In pyrimidine metabolism, the ring structure can be cleaved open. As a result, amino acids and ammonia (and some other compounds) are formed. So, both amino acids and pyrimidines can contribute to the ammonia pool of the body. Ammonia enters the urea cycle and is excreted as -not suprisingly- urea. BUN levels reflect this.
On the other hand, purine ring cannot be cleaved open, so its fate is different. However, ammonia is also formed as a result of purine catabolism (this is what adenosine deaminase does). So, purine catabolism will also contribute to ammonia pool. Ultimately, purines are converted into uric acid and excreted.
The bottom line is:
1) Both purine and pyrimidine metabolism yield ammonia, which is converted into urea
2) Only purine metabolism yields uric acid