Cast Iron vs Steel

What is the advantage of a steel skillet over a cast iron one? I currently use cast iron for most everything and am curious what I might be missing.

Pan sauces made with wine, vinegar, or any other acid are better in stainless steel. If you put any acid in cast-iron, you are harming your seasoning, and leeching iron into your food. This will affect the taste of your sauces, I find pan sauces taste metallic when made in cast iron.

Stainless steel also heats up and cools down much faster than cast-iron. This is great when you need quick heat, or fine control of your heat. You can also plunge a piping hot stainless pan into an ice-bath without cracking it in half.

If there's an advantage to getting a steel skillet as well, what would be recommended?

Go with a a bonded stainless-steel pan with an aluminum core. The most well known manufacturer is All-Clad. The stainless steel exterior is great due to it's non reactivity, you can literally put anything in it. The aluminum core distributes the heat much more quickly and evenly, minimizing hot-spots.

Is a steel skillet good for cooking omelettes?

Not in my opinion. I go with a non-stick pan every time.

I have to recommend sticking with a nonstick pan for eggs. There's simply nothing better, although well seasoned cast iron comes awful close. If you're spending more than $20 for a nonstick egg pan, you're doing it wrong. You don't need Calphalon, or any other big name for a good nonstick pan. Go to a restaurant supply store if you can and buy a cheap one there. With care it should last you 2-5 years depending on use. I found my current one at a Bed Bath & Beyond.

I agree with hobodave's answers, and let me add one more thing. A well made stainless pan will generally heat more evenly than cast iron. There is a myth about cast iron that it heats evenly, but it simply isn't so. It holds a lot of heat, which is a big benefit, but unless you move it around on the burner, there will be definite hot and cold spots corresponding to your burner pattern. (And I say this as a devoted but honest lover of cast iron, who uses it for almost everything I cook).

While yes, it is easier to cook on a non-stick pan, I have a different solution. I use a seasoned carbon steel wok to cook my eggs and it is my favorite way (besides sous vide) due to the ability to keep the egg contained within the oil.

As long as you put at least a Tablespoon of oil in the bottom of the wok, you can drop the eggs right into the center of the oil and it won't stick. Let them sit for a moment on the bottom to regain the lost heat. I then use a metal whisk to both stir up the eggs and scrape them from the center of the wok. Just don't get overzealous with it or you will end up with egg all over your wok.

This also works amazingly for omelettes since you can just leave the egg flat in the oil and then flip it. If you want to get it wider, try carefully moving the wok in a horizontal circular motion to spread the egg within the oil.

I have had bad experiences with teflon getting into my food in the past and I avoid it at all costs (even at low temps). But that is more personal preference, it can work well if handled properly.