Advantages of a 3/3 multi-class:

• More cantrips

• More known spells than sorcerer (6). [Same number of spells as a wizard (6) with the same starting INT]

• Increased spell selection over sorcerer (6)

• More sorcery points than wizard (6).

• Ritual spells (not available as a sorcerer without a feat)

• Metamagic can be applied to spells being cast via familiar

Disadvantages of a 3/3 multi-class:

• Multi Ability Dependant (MAD, requires Int and Cha, on top of Con and Dex).

• Does not know 3rd level spells (7th level characters know 4th level spells)

• Know less spells than a Wizard (6)

• Fewer feats/ASI (although a 4/4 has the same ASI as a single class 8)

• Fewer sorcery points than sorcerer (6)

• Delayed feature progression in both classes.

In my opinion (as a player with 2 years exp and a DM with 1 year exp), the MADness and lack of 3rd (and at the next level, 4th) level spells makes this a sub-optimal character.

Additionally, you are short a ASI/feat for a level, and will be less effective with your sorcery points because you only have 3.

It won't be so weak that you can't play it, but you will feel less powerful than the other dedicated casters (and maybe the half casters as well) in your group. if you are trying to build a strong character, you are better off going full in either class. Sorcerer will give you more power, but less options, while wizard will give you more options, but slightly less power.

You might consider only taking a 1-2 level dip into wizard, and mainlining sorcerer instead. Then taking ritual spells, and spells like mage armor and shield that don't require INT to cast. This will get you the increased flexibility of a wizard, but eliminate some of the MADness and let you get that ASI right away (you still need 13 INT to multi-class). It will also only delay your spell progression by 2 levels, instead of 3. You will have less pure power than a pure sorcerer, but significantly more flexibility. However, if the ritual spells are the only thing you are looking for, then taking ritual caster will be much more efficient than 1-2 levels of wizard.

In general, multi-classing full casters is a bad idea (even those who share a spellcasting ability). Every level you don't take in your main class is a level longer before you get spells for a slot that you already have. On a pure damage comparison, casting a 1st level spell with a 3rd level slot will do less damage than casting a 3rd level spell with that same slot. And that speaks nothing to the utility of the higher level spells. One or two level dips into another class are occasionally worth it, depending on how front loaded the class is, but any more than that and you start significantly delaying your spell progression.

Yes, you get more cantrips for multi-classing, but you're only going to use one or two most of the time, and by 6th level, you have 10 spell slots. You will probably only have 15 rounds of combat in a day, so you won't need your cantrips that much.

To directly answer your question (is there a new ability that the multi-class gets that neither original class gets):

No (ish).

There is no novel feature (like your examples of action surged spells or spellcasting in armor, although you could take feats to be able to do that as a pure wizard) that you get from multi-classing these two.

However, as per this question, metamagic can be used on non-socerer spells, so you can get new combinations by modifying your wizard spells. For example, if you took Protection from Good and Evil as a wizard spell (not available as a sorcerer), you could extend it using metamagic to have a 30 ft range. While this is technically a new feature, it is very situational, and probably not worth the multi-class.

You only have one pool of slots

You cast whatever spells you can cast, in whatever manner you cast them. They use slots, which are a generic resource independent of your specific spellcasting class. There aren't 'Wizard' or 'Cleric' slots, for example - just 'spell slots'. So your PC might cast only Eldritch Knight spells with all their slots or only Wizard spells with all their slots or some combination of both.

So, yes, you should ignore the class tables and just look at the table for multiclass spellcasters.