In the case you describe, you are not really jamming a frequency as much as jamming a protocol. Wifi has a well agreed upon way of scheduling use of the frequency, including how to "start a conversation" for lack of a better colloquial term. You can think of this like the start of a sentence. If you don't get to the end of the sequence, no data is transmitted. The deauthentication attack takes advantage of the fact that the deauthentication packet is not properly encrypted, so it can be spoofed. Whenever the attacker sees an authentication packet from a client to the wifi router, the attacker generates a deauthentication packet and transmits it before the client can finish its sentence.
In human terms, this would be something like this.
All kidding aside, we have a device that does this to human beings, making it so it is extremely hard to speak. It operates on the same principles. It turns out that the human body has a remarkable set of feedback loops designed to ensure we speak clearly. If you provide someone a copy of their own speech, delayed by a few milliseconds (as is being done by the cellphone and earbuds in the above video), it has a remarkable effect on the human voice. The mind thinks it is stuttering, and tries to correct for it. The result is well... just see the video.
When we start talking about "jamming a frequency," what we really mean is denying the ability to use that frequency for anything useful. This is slightly different than the above cases. In the above cases, we had a specific target individual that we were looking to jam, and we jammed them. To jam a frequency, you want to generate enough noise on that frequency range to ensure that nobody can use it (within reason). Typically, this merely involves an emitter which operates on the correct ranges.
White noise generators for babies are designed to decrease the "signal to noise ratio" (SNR) of any sounds the parents might make. The baby has a harder time distinguishing the bumps from the random content of the noise, so they don't flag it as "interesting" and sleep through the night.
SWAT teams are known to use bright pulsing strobe lights to "jam" the vision of anyone who looks in that direction.
LRAD is a sound device which "jams" a range of sounds so intensely that you can't even be in the area.
Size matters. It takes a lot more power to jam a SPY-1 military radar which weighs in at 6000000W of outputted power than it takes to jam an itty bitty Wifi router which tops out at 0.5W. (assuming they were operating on the same ranges.)
The difference between these examples of jamming a part of the spectrum and the above examples of jamming a protocol is that, when you jam a protocol, you are relying on them to follow their rules of engagement. If they ignored them, the jamming would not work. For example, if both parties decided they were going to ignore deauthentication packets (in violation of 802.11, of course), they would be able to communicate just fine.
The tradeoff is easy: the more targeted your jamming approach is, the less power it takes to do it. You can jam a Wifi signal with deauthentication packets from your laptop without breaking a sweat. If you chose a less focused jamming technique, the sender and reciever may agree to consider you to be a "noise source" and use the beam shaping techniques in 802.11n and so forth to massively decrease the effectiveness of your jamming. To be effective, you may need to output many times the power a laptop Wifi card is intended to output!