Mdthbs

If a climber completes a route without using their protection that is called sending.

What is the term if one completes the route but had to depend on the rope either for a rest or for a fall?

Let's say using the rope lowers the climber enough that they still have to complete all moves. In other words all moves have been done properly, but the ascent wasn't clean.

In aid climbing this is still sending since hanging on protection is part of the game (as @StrongBad points out, hanging on the rope is a little different than hanging on protection.)

In sport climbing, however, there is no specific word or phrase. I think the most common ways to describe it are:

Projecting

As in "I'm projecting Era Vella 9a." This doesn't capture the fact that you've done all the moves, but most sport climbers are focused on the clean send, and so a large part of projecting involves doing all the moves in sequence with the occasional hang.

Worked through all the moves

As in "I worked through all the moves on Predator 5.13b." This could mean that you rappelled in to try each move, but in some groups this will be understood as ground-up climbing.

...but not clean.

As in "I climbed my first 5.8 today, but not clean. I had to take once after the crux." This will usually be followed by rude people reminding you that you didn't "actually send", and nice people reminding you that you should climb however you want as long as its safe and fun.

Edit: I changed the example here from "sent" to "climbed" because after thinking more about Adam's comment, I think there is a whole connotation involved with "sending" that some climbers may not like appropriated.

I think it's important to realize, though, that not officially "sending" is no less impressive or cool than sending. My experience in sport climbing circles is that no matter how hard you send or how many times you hang, other climbers will be stoked for you if you're trying hard and enjoying yourself.

Without falling, wouldn’t it be hangdog, see https://en.m.wikipedia.org/wiki/Glossary_of_climbing_terms#hangdog

With falling, I also would go for projecting.

(Speaking of Sports climbing!)

I cannot comment @vladimir F
Sorry the red point, while coming from German and especially from German climber Kurt Albert, does not come from the Elbsandsteingebirge but from Frankenjura (a Limestone region)! See https://en.m.wikipedia.org/wiki/Redpoint_(climbing)

That is a free climb (the outcome). The term you may use is All Free (AF). It is much more established in German (Alles Frei). Remember, that RP (Rot Punkt) also comes from German. It (AF) makes more sense in traditional sandstone climbing in Elbsandstein, in multipitch clibing and in alpine climbing though. Check, for example, this book, I quote "The AF (All Free) style was developed in the Alps as the first recognized sport style of free climbing , where the climber uses only natural for advancement and and overcomes the route with his/her own strength."

I think the general progression I use is climbed, led, freed, on sighted, and flashed. I might break freed into red point and pink point if the route or area makes if ambiguous. I generally break led into a lot of terms like groveled, hang dogged, whimpered, depending on how ugly I was. Terms like projecting to me mean you are working towards freeing the route, but if I led a route with a single fall and have no plans to go back, I would describe that as projecting.