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:
inclined; disposed; given; prone:
too apt to slander others.
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..."