The princess cut diamond engagement ring offers a wide range of benefits, including:

1. Brilliance: Princess cut diamonds are known for their brilliance and fire, which is due to their faceting pattern. The princess cut has many small facets that reflect light back to the viewer, creating a lot of sparkle and shine.
2. Modern and versatile style: The square or rectangular shape of a princess cut diamond gives it a modern and contemporary look that many people love. Additionally, because the princess cut is so versatile, it can be set in a variety of settings and paired with different types of wedding bands.
3. Maximizes carat weight: The princess cut diamond is designed to maximize carat weight, which means that the diamond will appear larger than other diamond shapes of the same carat weight. This can be an advantage for someone who wants a bigger-looking diamond without having to spend as much money.
4. Durable: Because the princess cut has pointed corners, it can be less likely to chip or break compared to other diamond shapes like the round brilliant cut.

Overall, a princess cut diamond engagement ring can be a great choice for someone who wants a modern and versatile diamond shape that is both brilliant and durable.

While both are very popular, there are a few unique differences between round and princess cut diamond engagement rings:

1. Shape: The most obvious difference between these two diamond cuts is their shape. The round cut has a circular shape, while the princess cut is square or rectangular with pointed corners. The round cut is a classic and traditional shape, while the princess cut has a modern and contemporary look.
2. Brilliance: The faceting patterns of the two cuts differ, which affects their brilliance and sparkle. The round cut has a higher number of facets and is designed to maximize brilliance and fire. The princess cut has fewer facets and is designed to maximize the diamond's carat weight and brilliance, but it may not have as much fire as the round cut.
3. Price: The round cut is the most popular and in-demand diamond shape, which means it is often priced higher than the princess cut. The princess cut, on the other hand, is a more affordable option, as it maximizes the diamond's weight and therefore can appear larger for its carat weight.
4. Durability: The pointed corners of a princess cut diamond can make it more susceptible to chipping or breaking, while the rounded edges of a round cut diamond make it less likely to suffer damage.

Overall, both the round and princess cut diamond engagement rings have their own unique characteristics and advantages. The choice between the two ultimately depends on personal preference and style, as well as budget and the specific qualities of the diamond being considered.

Compared to some other diamond cuts, the princess cut can be a more affordable option, as it is designed to maximize the diamond's weight and therefore can appear larger for its carat weight. Additionally, princess cut diamonds are more readily available than some other cuts, which can also help keep the price lower.

However, the cost of a princess cut diamond engagement ring can still vary widely depending on the factors mentioned above. For example, a higher quality diamond with fewer inclusions or a larger carat weight will be more expensive than a lower quality or smaller diamond. The cost of the setting, whether it's made of gold, platinum, or another material, will also affect the price.