Should I clean the engine area to remove years of built up grease and dirt? What are the benefits?
Aside from the general desire for cleanliness, there are couple things that come to mind:
With clean engine it is easier to see if anything is out of order, such as fluid leak. This, potentially, can have dire consequences, if not caught in time as, according to U.S. NFPA report, 2/3 of vehicle fires are caused by mechanical or electrical failures.
In 1994-1998, roughly two-thirds (66%) of the highway vehicle fires were caused by some form of mechanical or electrical failure, such as part failures, short circuits or backfires. Part failures, leaks or breaks caused one-fifth (19%) of the passenger road vehicle fires. Short circuits or ground faults caused 18% of the fires in these vehicles, 10% were caused by backfires and 7% resulted from other electrical failures.
And while distribution of causes may differ by country (in UK, for instance, vast majority of vehicle fires are caused by arson, according to annual governmental Fire Statistics reports, compared to less than a fifth in the U.S.), significant portion of accidental fires are still caused by mechanical failures, and fluid leaks onto hot exhaust, or combined with electrical shorts and other similar causes, are a serious concern.
In the above report, annual averages had shown that even general lack of maintenance was attributing cause to slightly more fires than as result of damage in collisions or overturns.
Dirt, grime, oil, road salt, and other contaminants lead to premature deterioration of such engine components, as hoses, electrical wiring, sensitive electronics (ECU, sensors, etc.) and so forth. Some of this, again, can lead to issues presented in the first item of this list.
Having a clean engine bay is really nice when you need to work on the engine, and it helps identifying leaks. There are a few things to be careful about when cleaning though.
Pressurized water can get into electrical connections and cause shorts.
Water causes corrosion.
Cleaning old degraded plastic connectors and shields can cause them to fall apart.
If you clean it, make sure you disconnect the battery, let the engine cool, and then clean everything. Afterwards use compressed air to dry out all the electrical connections and blow the pooled water out of all the crevices (especially fuse boxes), then let it dry for a few hours (best to do on a warm day). Connect the battery and drive it for a good hour to get it hot and finish drying out.
On the downsides, when buying an used car, I'd be suspicious of vehicles with overly clean engine compartments (even though some dealers would do it as a matter of procedure to all cars they resale). This is usually an indication there is a serious leak somewhere which will almost surely be expensive to repair. This is more valid when purchasing a car from a private individual instead of going through a dealer.
So, you might not want to scrub it shiny just before reselling your car, but as a matter of maintenance, it positively won't hurt.