But is saving money better than using it at all? I'm not broke, just cheap.
As above, the reason I posted what I do is because neither are necessary, and both can actually be worse for the tank than better.
For example, consider the tannins that you are removing with the carbon actually can do your tank good. They are large complex organic chemicals that bind up with heavy metals easily. This does two good things. One, a bound-up heavy metal is significantly less toxic for fish than an un-bound one. And, the amount of heavy metals in the water that is safe for some fish, is much less than what is safe for humans. And, water companies only have to make water that is safe for humans. Copper, for example, is typically lethal at fish around 1/100th the concentration it is needed for humans to taste copper in their water -- and the concentration when humans can taste the copper still isn't lethal to humans. Now, most water doesn't have a lot of copper in it, but it is one example. And, having tannins in the water is just a nice extra safety net to bind up a stray heavy metal or three that may get into your water. Two, a bound-up heavy metal is easier for plants to take in when they need micronutrients. It is kind of a neat cycle -- a plant dies, releases its tannins, the tannins bind up micronutrients so that the alive plants can take them in easier. In addition, some fish spawning can be triggered by tannins. So, all in all, I just don't see why you would want to take them out.
As above, the zeolite that absorbs ammonia and other nitrogenous waste, does almost nothing but starve the cycling bacteria and live plants you may have in your tank. Most aquatic plants prefer to take in ammonia as their source of nitrogen, and if you don't have plants, then most of us culture a colony of ammonia and nitrite consuming bacteria. If the plants or bacteria don't get food, they die. And, then when all the zeolite is fully saturated (a process that happens very quickly, 24-48 hours typically), they isn't anything alive to consume that ammonia and protect your fish from ammonia poisoning.
So, as I said, to really save money, I wouldn't use either.
In fact, I take extra work to remove carbon from my filters. The penguin filter pads for my HOB come with carbon inside them. I take a utility knife and cut open the pads and shake as much carbon out of the pad as I can. As I wrote above, I don't want carbon taking up good things in my water, and since I don't have to worry about replacing the carbon, a filter lasts around 6 to 12 months for me.
Finally, it is a good idea, as other posters wrote, to have some carbon and maybe even zeolite around for emergencies. It is nice to put carbon back into the filter when finishing a course of medication. Water changes will dilute the medication, but won't ever get all of it, but the carbon can adsorb the rest. And, if you live in an area where you may have long power outages, it may be nice to have zeolite on hand to be able to hand in a stocking in your tank to minimize ammonia if needed. But, neither is needed at all for normal operation. And, that really saves you money.