It’s like poetry or whatever but the Knight of the Laughing Tree and the Knight of Tears as fascinating parallels of and inverses to one another.
In both cases, the goal of the mystery knight is to make an objective point, without the details of their respective identities muddling or neutralizing the message. Indeed, one potential advantage of an individual donning the disguise of a mystery knight is the ability to send such a message to the competitors and/or attendees of any particular event, pursuing a goal otherwise unobtainable or complicated by their persons. Yet where other mystery knights may have wished simply to convey their individual worthiness to compete irrespective of their identities (as, perhaps, Baelon Targaryen did when he tilted as the “Silver Fool” to win his spurs at Old Oak, or as Jonquil Darke did when she entered the War for the White Cloaks as the “Serpent in Scarlet”), Prince Aemon and Lyanna Stark were not interested (or, at least, entirely interested) merely in jousting for the sake of jousting; their victories in the tilt mattered less than their motivations for doing so.
Instead, for both Aemon and Lyanna, their points attempted to correct wrongs which inextricably linked the personal and the political. Aemon’s personal love for his sister (complicated as it might have been by his devotion to the vows of the Kingsguard) was clear, and on those grounds alone he might well have resented the preferment of his brother’s mistress over his beloved sister. Yet the insult did not end at mere familial (or even potentially romantic) closeness between Aemon and Naerys. By publicly attempting to snub the present queen for his current mistress, Aegon IV was declaring to the assembled courtiers and aristocrats that his wife - the only woman he could, at least openly and officially, have a publicly approved romantic/sexual relationship with - was not the fairest woman in the land (or at least among the tourney attendees). Beyond being a major breach of courtly etiquette, such a move could suggest greater ambitions on the part of the king to remove Naerys from her place as queen - no empty threat, when Aegon IV had (likely) already used Morghil Hastwyck as a proxy to accuse Naerys of adultery and when the Brackens had schemed to replace Naerys with Barba. Aemon, himself the champion for Naerys against Morghil and one of the voices to call for Barba’s dismissal, would thus again take up the defense of his sister-queen to an insult at once personal and political.
Likewise, Lyanna framed her rescue of Howland from the bullying squires in terms at once familial and feudal. Howland was, so Lyanna declared him, “my father’s man”, the most fundamental expression of the political order which obligated her, as a member of the liege family, to protect the liege’s vassal with her own power. Just as Lyanna cared for Howland herself following the attack (“t[aking] him back to her lair to clean his cuts and bind them up with linen”, in the tale told by Meera Reed) and welcomed Howland into the bosom of the Stark household at Harrenhal - again emphasizing the personal responsibility of her as a Stark to look after a Reed of Greywater Watch - so Lyanna recognized her role as redeeming the honor of young Howland. It was her duty as a resident Stark to personally take up arms against the master of those squires and defeat them (albeit in a play-combat context), much as Lord Rickard would have been expected to do should anyone have made war on any of his vassals. The insult to Howland had triggered Lyanna’s personal obligation to him as a Stark; here was her opportunity to do as her Stark ancestors had done for generations, caring for and defending the people under Winterfell’s protection.
Yet Aemon and Lyanna diverged in the means by which they conveyed their respective messages, placing them on opposite ends of a symbolism spectrum. For Aemon, the proper moniker to demonstrate his point was the Knight of Tears. Even without the specifics on his shield’s device, the designation conveyed Aemon’s sense of grief at the humiliation of his sister. While the disguise may have subtly recalled the history between the siblings - when Aemon had, so the songs relate, wept to see his sister wedded to their brother - its main purpose was, I think, to communicate the objective shame of the king’s proposed action. If the king himself would forget (or actively refuse) his social duty to his lady wife, Aemon would remind him - not as his brother (whom then-Prince Aegon had ignored in their quarrel at Aegon’s wedding to Naerys), nor as his Kingsguard (since Aemon would, at least in his mind, owe the king his unquestioned loyalty), but as the representation of a chivalric ideal. To so great an insult, Aemon’s guise suggested, any true knight must surely weep - and only such a true knight could redeem the honor of a queen so disparaged by her king.
Lyanna, for her part, also sought to shame the men who (indirectly) humiliated the person she sought to defend, but in a way which more directly mocked the knights in question. The Blount, Haigh, and Frey knights she challenged may not have directly or publicly insulted Howland Reed as Aegon IV attempted to do to Naerys, but their squires’ harassment of Howland reflected none too well on the knights themselves as ostensible instructors of honor and chivalry. The laughing weirwood thus mocked the pretensions of both the knights and their squires to the outward appearance of chivalry. Where the squires had failed in honor by attacking a young and much smaller man, their knights would fail in the public demonstration of knighthood, being roundly defeated by the Knight of the Laughing Tree. This mystery knight, and openly bearing the symbol of a non-Andal, non-Seven-worshiping land and people (and thus without inherent ties to the Andal tradition of knighthood), would humble the knights whose squires had acted so unchivalrously. That Lyanna was of course no knight herself only underlined the joke intrinsic to the choice of device: she could not be the chivalric ideal as Prince Aemon (himself one of the most publicly celebrated knights of all time) had been as the Knight of Tears, but she could secretly advertise that these knights would be unhorsed by a teenage girl with no formal knightly training.
In turn, both Aemon and Lyanna found (or likely found, in Aemon’s case) in the outcomes of their tourneys reversals of their chosen disguises. Aemon might have symbolically wept to see Naerys so insulted by their brother, but in emerging victorious in the tilt, Aemon perhaps restored a smile to Naerys’ face. In being presented with the crown of the queen of love and beauty, Naerys had been publicly acknowledged as the fairest woman in the land (at least in the context of the tourney). Too, that it was a mystery knight who presented her this honor (supposing Aemon was still disguised when he did so) may have only underlined the triumph in Naerys’ mind; to all onlookers, this was not a brother playing favorites against their loathed eldest sibling, but a true representation of knighthood defending the position of the queen. Aegon IV would not cease his attempts to humiliate and undermine his wife after this event, but in at least this instance, the queen and not the king’s mistress (or the king himself) would have the last laugh, thanks (with no small sense of irony) to the Knight of Tears.
By contrast, while Lyanna had chosen a laughing device for her tourney stunt, the end of the tourney of Harrenhal was “the moment when all the smiles died”. If Lyanna had succeeding in (literally) beating honor into the knights whose squires had so abused Howland Reed, she then found herself on the opposite side of the knightly dynamic, the unexpected recipient of the crown of the queen of love and beauty at the hands of Rhaegar Targaryen. Where his great-great-great-great-great-granduncle had refuted an attempt to honor the king’s mistress and instead acknowledged the queen as the rightful queen of love and beauty, Rhaegar likely seemed, to the attendees of Harrenhal, to do precisely the opposite - grossly publicly insulting his own present wife (and the future queen) for the sake of a would-be mistress. Not for Lyanna the power or desire of a Prince Aemon, to follow through on crowning a queen of love and beauty, and so not for Lyanna the ultimately happy outcome promised by her weirwood’s laughing face; hers would be that “sadder story” alluded to by Meera Reed, one more full of tears than Aemon’s tourney victory.