As a minor point, it may not be best to not call Æthelred a Saxon. Yes, he was a member of the house of Wessex, but after Æthelstan all of England is united and you can really start talking about Anglo-Saxons as a real group and identity in Britain.

There would have been many differences!

The easiest to identify would be the language that they spoke. Whilst the Saxon court probably spoke a mixture of Saxon-Germanic-Anglic language, the Plantagenet language spoke a well defined blend of late Anglo-Saxon Germanic (or, more easily: early English) and early French.

We'll also see a massive difference in bureaucratic organisation. Whilst the Saxons depended on a rather limited and localized system of fief control and bureaucracy, the subsequent Norman and Plantagenet kings were centralists at heart. We can see effords to unify and nationalize the bureaucracy starting with William The Conqueror, all the way up to Henry III. Laws and courts were nationalized, and under King Richard (and his infamous brother John) court procedures became more and more a matter of the crown. King John especially had a personal interest in the conduct of law courts, and when travelling, would often personally intervene in local judicial procedures. In addition, attempts to nationalize and streamline taxation and troop mustering were made by William the Conqueror, Henry I and Henry III. Indeed, it is during Henry III's reign that we first see the term 'Parliament' being used for large gatherings at court, and Parliaments were held a couple of times during his reign. These parliaments were organised to gather support for levying special taxes (mostly on movable assets) to fund specific projects of the Crown. Currency was also reformed under his rule. Centralized rule for kings of this period was easily imaginable as a more effective way of government however; as their empire was far larger than that of Saxon kings like Æthelred. Æthelred controled most of Southern England, whilst Henry III was lord of England, part of Scotland and even parts of France.

In more day-to-day terms: Henry III traveled less than previous kings, and stayed at his palaces and castles for longer periods at a time, often expanding or beautifying them when he was there.

In contrast, Saxon kings moved around a lot. Looking at the example of Æthelred, we can see why he had to. The Danes were a constant thread on his coasts, and he had to combat numerous raids and invasions during his reign.

This is not to say that the Saxons were disorganized. Indeed, it was Æthelred who revamped the system of law and courts in Wessex at the time, inventing a system wherein 12 Thanes supported by a royally appointed Reeve had the responsibility to administer justice to "Charge-Laden" men.

So; to summarize: the main difference in conduct can be found in the way that the Kings conducted their affairs. Whilst Æthelred generally favored leaving local matters to local officers (who obeyed royal law under oath), Plantagenet kings generally took matters into their own hands, attempting (if sometimes unsuccesfully) to centralize government in England and their other territories. Both had their reasons: Æthelred benefited from a more local and flexible system because he was a king on the move, dealing with issues as they arose in his domain. Henry III lived in relatively peaceful times (right after a bloody civil war) and thus preferd, and could afford to, conduct kingship from one or a few locations.

As for my sources: I recommend Michael Pye's The Edge of the World. How the North Sea Made Us Who We Are if you want to know more about early medieval government, life and economy around the North Sea. Also, the Plantagenets: Warrior Kings and Queens that Made England by Dan Jones is an excellent read, if a bit popular sometimes. I also used this website detailing early english law and systems of government. Great page if you want to find sources on early medieval law.