What's the difference between un- adjectives and -less adjectives?
According to Michael Quinion, Ologies and Isms: Word Beginnings and Endings (2002), -less words fall into two main categories: words where the -less form indicates simply "without" (as, for example, bottomless, childless, defenceless, and lawless), and words where the -less form "indicate[s] something that is unaffected by the action of the verb, or some failure or inability to carry out that action" (as, for example, dauntless, quenchless, relentless, tireless).
In contrast, Quinion says, the prefix un- usually lends the simple meaning "not" to the resulting word; but in some instances it "often has a stronger and less neutral force than just negation (so it is not equivalent to NON-): unkind can mean active cruelty rather than simple lack of kindness; to say someone is un-American can imply an active antagonism to American ways. Furthermore, with verbs, Quinion says, un- "usually has the sense of reversing some state" (as, for example, unburden, unlock, unsettle, untie).
So on a basic level, the split between -less words and un- words is the difference between "without" and "not." But in addition, Quinion suggests, some -less words imply a resistance to an embedded verb, and some un- words either express a reversal of an embedded verb or go beyond mere negation to suggest an active opposition to an embedded adjective.
In the original examples listed, there is not only a difference in the prefixes, but also the suffixes. This goes along with FumbleFingers' comment that "-less normally attaches to nouns, but un- primarily attaches to adjectives/adverbs." And to further complicate the matter, un- can also attach to verbs (usually with a slightly different meaning), and there are some adjectives that derive from participles of these verbs, so we get contrasts words like undone that can have two distinct meanings ("never done" or "once done, now reversed").
This is relevant because usually, you can't actually attach these affixes to the same stem; three of your examples use a noun X and a related but not identical adjective Y as the stems for the pair of words. (The exception is dauntless and undaunted, which the dictionaries that I have consulted say are both from the verb to daunt). Differences in meaning may already exist at this stage (talent n. vs. talented adj., luck n. vs. lucky adj., hap n. vs. happy adj.).
As others have mentioned, X-less can be approximately glossed as "without X," and un-Y can be approximately glossed as "not Y."
So in this answer, I'm going to discuss how the suffixes used may affect the meaning, and how generalizable the meaning of the suffixes is. I'm sorry if this is somewhat tangential to your original question; however, I thought I'd mention it since I don't see it discussed in any of the other answers.
- luckless and unlucky
In this case, we have the suffix -y used to form an adjective meaning "characterized by (having) X" from an noun X. Adding the prefix "un-" to this adjective would according to my gloss result in the overall meaning unlucky = "not characterized by (having) luck"; in this case, the scope of the negation in actual usage is always narrower "characterized by not having luck." (Sven Yarg's answer describes this better.)
- hapless and unhappy
Here we also have the same suffix -y. But, the meaning of the adjective happy has actually evolved further than the original noun. It is not accurate to define it as "characterized by (having) hap." The word hap (which is now archaic) had a primary meaning of "good luck" or "prosperity." While the adjective happy can still sometimes mean fortunate (applied to an event or a person), it is often used to mean simply that someone is pleased or contented.
This difference in meaning is reflected in the derived words hapless (which Merriam Webster defines as "having no luck : very unfortunate") and unhappy (according to Merriam Webster, "sad, depressed, or disappointed : not happy").
Happiness is a related noun that is more equivalent in meaning to happy (because it is derived from the adjective). And "without happiness" seems pretty much a synonym to "unhappy." But, there is no such word as *happinessless (this is part of a more general trend where we generally can't apply the suffix -less to nouns ending in -ness).
-less generally means without, lacking in, or an inability to do; un- typically means not, or the reversal or opposite of some state. However, in the common vernacular, I think the distinction only applies on a case-by-case basis. For example, 'unlimited' and 'limitless' both mean sans limits, however 'stainless' and 'unstained' have come to mean two different things (the first implies that it does not stain ever, whereas unstained suggests that something has yet to be stained.)