Attributes:
Strength, Agility, Endurance, Intelligence, Willpower:
With the old system you had multiple attributes affecting health, stamina, and magicka. Each of the attributes listed above played a different role in determining your amount of "life force". The new system really just combines the effects into one simple classification. Magicka, Health, and Fatigue each contribute to your base amount (fatigue also represents carry weight) and also your amount of regeneration.
You also see Strength, and Willpower show up in the different perk trees. Most weapon skills allow you to increase how much damage you do with said weapon (the same was said for strength) and Willpower shows up in the different magic trees as there are perks to increase magicka regeneration, resistance, and absorption.
Luck:
Luck still persists throughout the game, however it's now represented it its true form (that being percentages). You find that the lockpicking tree has perks that give you a greater chance of finding gold, and treasure. Some trees like pickpocketing give you a higher chance of pickpocketing, and many weapon tries give you a chance to crit, paralyze, stagger, or damage avoidance. It was always said that luck affected you in everything you did, and that still persists within Skyrim.
Speed:
Gone, dead, M.I.A, and good riddance.
Personality:
Pretty much dead. The perks in the speech tree are pretty much all flat rates. There is no haggling, and persuasion attempts exist, but are numbered.
Spellmaking:
To be fair you somewhat make spells. While you cannot ultimately define the parameters. You do have the ability to influence the amount of spells damage through perks, and by dual casting. While there isn't as much flexibility as previous games it allows for spells to stay within the parameters set the by DEVELOPERS. You cannot freely change the damage value of weapons, and in a sense the removal of spellmaking was to make it more balanced. However, while this does ultimately hurt spells in a negative way it actually forces you to use the different magic schools, rather than focusing on one. I've played a s mage on master and it's a pain, however the same is true for my warrior. Basically any pure class is at a disadvantage on master because you need multipliers (like sneak, or magic) to help you defeat enemies.
While spellmaking doesn't have the flexibility of older games, it still allows you to influence the damage of spells. The game has also introduced (to my knowledge) one of the more fantasy, lore based setups for acquiring summons. The atronach forge is an interesting tool, and while it doesn't replace spellmaking it could be a much more interesting addition, especially if modders add a whole new set of summons through the forge.
Spells:
Chameleon:
Is part of the Sneak tree, as you can eventually walk around undetected when you pump enough points into sneak. One could argue that they would like a spell that allows you to remain undetected, however you still have invisibility and the ability to actually interact with objects should be for those well versed in sneaking.
Damage health/magicka/fatigue:
Part of the destruction school of magic as the fire also has a damage health effect, frost has a damage fatigue, and shock has a damage magicka. I honestly like this system better as fire would do lingering damage, cold would sap your energy, and also your dexterity. Lightning would... kill you? Magic doesn't exist in the real world so drawing correlations between the two is hard.
On top of that you also have the same effects with poisons.
Weakness to Frost, Shock, Fire:
Seen in poisons as well. Although this is somewhat useless as applying the poison can be difficult for pure mages.
Feather:
Perk in the lockpicking that gives a permanent 100 point boost to your carry weight. Ultimately isn't the same, but is actually more useful in the long run as it doesn't fade.
Damage armor:
Seen in a shout, along with damage health.
Damage Skill:
Not really seen as the skill itself has very little, to no influence on abilities, rather what you spend your points on influences your abilities. The closest thing to a damage skill would be disarm which puts the enemy on an uneven playing field.
Open:
Not really seen as a spell. The tower stone has the ability to open locks, however the minigame is easy enough and one could argue that the lockpicking tree itself renders open useless.
Absorb health:
Only seen in enchantments.
Magic shields:
Still seen in some form. Whether it be the wards, or altering the users flesh the ability to shield yourself from damage or magic still exists.
Levitation:
Gone. Allows for too much flexibility, adds the thread of game breaking problems, and would allow for the user to basically skip steps like going up 7,000 steps.
Summary:
I understand to some what I've said will seem like I'm grasping at straws. However, although the names of features have changed, the effects have not. Many of the spells, or features still persist in some form or another. Some play a reduced role and have since Oblivion. Some features have taken a back seat to player skill, and others have simply been taken out because they were either caused too many problems, or allowed for too much flexibility.
One could argue that the removal of some features allowed for less options, which is partially true. While some features have been removed, others have been replaced or changed. While some may prefer the older system, finding newer features to be useless, the same argument could be said but from the other side. Many older features were useless, while some of the newer features are more useful.
The most interesting idea interesting change is that the character is more fully realized. The character may not be simply represented by numbers dictating damage, carry weight, and success, they're represented better visually. Before when you were a master archer you merely did more damage, or had a higher chance of hitting a target. Now you have the ability to slow down time, move faster with an arrow drawn, and draw back a bow faster. If you're a sword and shield type player you have quicker reflexes with a shield that allows for you to essentially move faster than your opponent, you can also disarm and stagger your opponent.
While there are less weapons, each weapon type has unique capabilities. Swords can crit, Maces ignore armor, and Axes cause bleeding. Before weapons were really just different skins, and while this new system isn't super deep it allows for more interesting play.
Afterthoughts:
I already know hardly anyone will read this, nor will many respond. But my point in posting this thread is to hopefully show that the game isn't as shallow as many think. Your character feels, and looks, more like a warrior, mage, or archer than they did before. The problem is really that there is only so much numbers can represent, and I know people like numbers because they provide infinite depth, but they're somewhat dated. The combat in Skyrim is subtle, and to some they feel it is boring, or terrible. However there are a lot of subtle things that enhance the combat like time being slowed down, or different magic/weapons having different effects.