I think I know where you are going, so first I will answer from a practical percpective and then if you care to read some definitions.
First, the only difference between a cable from your wall and an antenna is the way the signal gets from the source (TV station) to your house. In the case of an antenna the TV signal is sent using radio wave through the air that is converted to electrical signals by the antenna, then that signal goes through the antenna wire to your TV. In the case of cable, the signal is send as an electrical signal from your cable provider to your home over a wire the entire way.
At the end of the wire, no matter if that is from an antenna or cable service, there is an electrical signal the needs to be converted to a TV picture and sound.
An analog signal in basic form can be thought of like waves, where the height and length of the waves represent the TV picture and sounds, in how the waves vary, these variations are infinate.
A digital signal only contains 2 states, that is, unlike the infinate variation analog can be, they only have 2 states they can be in, called 0 and 1 (low and high). By making a LOT of 0's and 1's they form a pattern that can, like the analog signal, be made to represent the picture and sound.
Now to your TV, if you have an older TV it only knows how to convert analog signals to a TV picture and sound, so with this kind of TV, no matter if the signal comes out of a cable from a cable company or an antenna wire from your antenna, it needs to be converted to something your TV can understand so it can reproduce the TV picture and sound.
So in both cases you need a converter of some kind.
That said, some cable companies still send an analog signal and thus still work on older TV's without a converter box, which adds to the confusion, so to make all things equal, we'll assume your cable provider has gone digital, if that is the case you need a converter to use the cable from the wall with an older analog TV.
OK, now to the quality, first of antenna vs cable. In an ideal world they can be equal, however we live in a very non-ideal world and antenna's will turn any form of radio waves regardless ouf source in to an electical signal, quality of TV received by an antenna can have many things disrupt them, lightening, other radio waves, physical objects between the TV station transmitter and your antenna, and a ton of other things the impact the quality of the picture and sound, most often seen as "snow" in the picture and pop's and clicks in the sound, you can also have a weak signal that can end up too weak for your TV to use.
Cable is much harder to disrupt, the TV signal travels in a special kind of wire that shields it from most external signals that would disrupt it, also the cable company when things work right has much more control over the strength of the signal that goes to your TV, there are no issues with buildings, trees, and within reason distance to make the signal weaker, so in most cases cable TV signals are more reliable and "steady" than those received on an antenna.
Cable costs money because they not only offer better quality in most cases, but more stations, without having to have those stations being close to your house. Broadcast stations have lower costs for their transmitters than cable companies with all that cable and the systems needed to drive the signal over all that cable. Broadcast TV can easily make a profit just selling commercial time to advertisers (companies), cable companies have more stuff to maintain and while they make money from commercials also on their basic channels, they also make money from consumers who value the better quality, far simpler use (no antenna to install and maintain for end users), as well as special programming, like commercial free stations, called premium channels that they charge extra if you want them, like for HBO or pay-per-view movies, sports, and other progamming.
So now back to the TV, older TV's as I mentioned can only deal with analog signals no matter how they get to the TV, cable or antenna.
Most newer TV's can deal with both analog and digital signals, again no matter how they get to the TV, you can buy a new TV that can receive digital broadcasts from you antenna and you can still have free TV, older TV's need a converter box that converts the digital to analog, they are real cheap and allow you to still get free TV from your antenna if you have stations in range.
Now for HDTV, no matter again how the signal is sent to your house just require a HD TV as long as the TV gets the signal. Now comes another confusing part, cable companies charge more for HD signals for a number of reasons, some technical and some just consumer desire, so to get HDTV from most cable providers you have to pay for that service. You can get over the air, digital, HDTV signals for free also, but the number of stations will be limited compared to cable.
Finally the advantage of digital over analog, with digital your TV, cable box or converter box only need enough difference in the signal to tell if it is getting a one or a zero, in addition to that there is all kinds of techie things your digital TV/box can do to fix to a degree the one's and zero's if they get messed up. Now on digital if your TV/box can't figure out the 1's and 0's you get nothing, no picture and no sound, so with digital it is all or nothing, you either get a high quality picture and sound or you get notta. With analog you can get weak signals and/or signals that have been distorted and still get a picture and sound, not good but something.
In the end, digital usally works out better, since if you get the signal, then you get as good a picture as possible with the TV you have, which is what most people prefer.
Finally the inputs to a TV, they come in 2 basic forms, signal input that comes from an antenna/cable that the TV must do two conversons on, the first is isolating the signal you want (tuning in the channel) from all of the others, and then it must convert that to a video signal for your picture and an audio signal for the sound. (Both analog and digital TV's have to do this for the "antenna/cable IN" input on the TV.
The other input most TV's these days have are Video and Audio inputs, in this case you need a box, like a cable box, to tune the station you want to watch and convert it to video and audio. In this case you hook the box to the cable/antenna to do the tuning and conversion, and plug that in to the TV's video and audio inputs.
This allows TV's to be universal, that is, they can work with any cable company type of signal or any over the air signal, since the TV does not have to isolate (tune) the desired station, or convert that signal to the proper audio and video, the box does that and the boxes are made to always send the same kind of video and audio to your TV.
Sorry there was no real short answer to that question...