The basic difference between the two is emotions it appeals to.
Fun is enjoyable. It causes pleasure - especially in active forms, as thrill, exhilaration, challenge, elation.
Interesting appeals to curiosity - learning, it's about things we want to know, see, learn, examine. Whatever reasons - be it for pleasure, or e.g. for professional interest or satisfying anxiety.
Since usually satisfying curiosity is pleasurable, these two are often correlated, but not always. A gruesome sight, say, bowels sticking out of a live person's ruptured abdomen, may be interesting, but definitely not fun. A secret document on enemy military movements will be interesting too, but definitely not fun. The pilot of a damaged airplane will definitely find the damage report interesting and absolutely not fun.
On the other hand, if you solved a hundred crosswords, solving one more may still be fun, but hardly interesting. Riding down a slide in entertainment park will be described as fun too - it may be interesting the first time, but then you're not curious about it any more, you just do it for fun. Being given unexpected presents is fun - but since you didn't expect them, you didn't have time to find that interesting.
In your examples, they are freely interchangeable without conveying different degrees of expectation (cf. apt definition 2 and likely definition 1 - both below). The main difference is that apt's more popular definition (def. 1 prone) influences the connotation toward conveying a sense of inherent history and habit1, while likely isn't as strong in connoting past experience, stressing mere present probability.
So "my dog is apt to run away if I don't close the gate" tends toward "my dog runs away most of the time when I leave the gate open", while "my dog is likely to run away [...]" sounds more like "dogs run away when they can, so my dog probably will". Note that this is a nuance, not a definite demarcation, so you can still use either in the exact same situation.
Dictionary.com entry for apt:
1.
inclined; disposed; given; prone:
too apt to slander others.
2. likely:
Am I apt to find him at home?
3. unusually intelligent; able to learn quickly and easily:
an apt pupil.
4. suited to the purpose or occasion; appropriate:
an apt metaphor; a few apt remarks on world peace.
Dictionary.com entry for likely:
1. probably or apparently destined (usually followed by an infinitive):
something not likely to happen.
2. seeming like truth, fact, or certainty; reasonably to be believed or expected; believable:
a likely story.
3. seeming to fulfill requirements or expectations; apparently suitable:
a likely place for a restaurant.
4. showing promise of achievement or excellence; promising:
a fine, likely young man.
I know you didn't ask for this part, but it hadn't struck me before that apt and likely have several definitions that are also quite similar to each other. And it does address the title!
Apt def. 3 is comparable to likely def. 4 (roughly promising), and mostly interchangeable, though less so than apt 2 and likely 1. Apt 3 is closer to talented (good at a specific objective, e.g. an apt learner, not usually a likely learner), where likely 4 is more generally positive (a likely young man, not usually an apt young man unless it's referring to his proficiency in a certain endeavor).
Apt 4 is comparable to likely 3 (roughly suitable), though they aren't quite interchangeable. Apt 4 is usually used for an action (an apt apology, but an apt location works too), where likely does not refer to an action (not a likely apology, but a likely location is fine). And where they overlap, there is a distinct difference. Likely 1 heavily influences likely 3 so that it connotes a favorable level of probability that the subject will suit (hence "apparently" suitable), where apt cuts to the chase and simply calls the subject suitable.
Note too that though I mentioned apt 1 earlier, it is not accurately replaced with likely, though as Jim notes in a comment, when something is prone to happen, its likelihood is increased.
Lastly, likely 2 is not semantically correlated with apt at all.
1: I found several discussions on these words by googling apt vs. likely, and I really liked the way grammar.com described apt: "The word apt typically suggests that the subject of the sentence has a natural tendency to enhance the probability of the outcome..."
Best Answer
You could say that a dry run is a type of rehearsal. Specifically, the type of rehearsal where you run through a show without stopping to actually work on issues, for the sake of testing where the ensemble's cohesiveness and preparedness is.
Rehearsal is a time set aside to rehearse, or, to work on a specific set of pieces, a show, or some other event. Sometimes, a rehearsal is involved in the sense of working on issues with the group, and addressing questions and concerns. On the opposite is a very hands-off type of rehearsal, which is the dry run. This is the rehearsal that can happen at specific determined checkpoints, to make sure that things are ready, or to assess all the issues at once before settling on a plan to address them in full before performance time.