Any answer (and thus any successful argument on your part) is dependent upon the ecology of the world as your DM envisions it. Per PHB page 65, a Druid is all about the ecology of the world, its balance, and against that which is unnatural, which includes aberrations.
The General Differences
Some of what makes any aberration or monstrosity "cooler" (your words) is the unnatural (be it demonic, alien, magical, etc) innate ability inherent in the creature. I'll offer two examples:
Monstrosity (Cockatrice, MM p. 42, CR 1/2) has the ability to turn things into stone. That is a powerful special effect. No
beast has one similar at that level. (See conditions: Petrify,
restrained, PHB 291). Compare to a CR 1/2 Ape (MM page 317). He can
make two fist attacks, but he can't turn you to stone.
Aberration (Gibbering Mouther, CR 2, MM p. 157) has the gibbering effect much like the Confusion spell, a fourth level spell. (PHB: p. 224) A Cave Bear is a beast of CR 2 with a non-trivial damage output (1d8 + 2d6 + 10) but no magical effect. A Giant Constrictor Snake does 2d8 +4 and may restrain a creature (lock it down), which is hard on a single creature. Confusion (a 4th level spell effect) at will is far more powerful and can have an impact on multiple creatures.
The examples are not exhaustive. Another example1 would be a monster that is invulnerable when it does not attack - this would be annoying on a monster, but on a PC it could significantly unbalance a campaign. (1 @NeilSlater pointed this out in a note).
An argument against this proposal is that beasts don't have special abilities as powerful as aberrations or monstrosities at similar CR levels.
Another argument against Aberrations is the specific enmity Druids have against such unnatural creatures.
An argument for this proposal is that some beasts do have magical abilities. See Flying Snake for example. (MM p. 322) Flying snake: real snakes don't fly; fly is a third level spell.
Monster Manual on Aberrations, Beasts and Monstrosities
Aberrations have innate magical abilities drawn from the
creature's alien mind rather than the mystical forces of the
world. (p. 6)
Monstrosities: Not truly natural, and almost never benign. (p. 7)
Beasts: Natural part of the fantasy ecology. Some have agical powers. (p. 7)
Revere nature above all, and get their powers from nature itself, or a
nature deity. (p. 64)
This is mostly arguments against. Natural is what a Monstrosity or Aberration is not.
But ... your DM may wish to pursue this with you
What does your DM consider to be natural, and a part of the ecology of the world? There are many Beasts in the Monster Manual that take our primary world animals and make them something 'magical' or 'fantastic' or 'unnatural' (pick the term you want to you use), even though they are well placed in your DM's secondary world. If there is a place in his world where Gibbering Mouthers are as common as elk are in ours, you have a stronger case than if there isn't.
If you want to sell the case for going beyond Beasts, I suggest that for the mechanical approach you compare the innate abilities of a similarly CR rated Monstrosities with various Beast's and see how the power stacks up.
Based on my brief review, the Monstrosity or Aberration tends to have a more powerful effects per CR. Perhaps, for Monstrosities, agree to a CR divided by two or CR divided by four, or a CR one to two steps down CR table2 as compared to Beast of similar CR that your Druid can transform into. (2Tables on page 112, Basic rules). Aberrations would be a harder sell in any case, even with a power reduction.
You may find a way to make it fit if the DM waives the RP element of what Druids are in the general context of D&D as applied to the specific ecology of the DM's world. You could argue that a Winter Wolf is a highly evolved wolf species that was influenced by the Weave (or magic as a native feature of a magical world) and could fit as Beast versus Monstrosity in terms of category (mechanics). This would still require balancing of special abilities.
For purposes of this discussion, I'll treat the table in the PHB as capping CR at 1 (Druid Level 8) for Wild Shape. This avoids the need to compare your Druid's related at will power (cantrip frostbite, 1d6 dmg to one target (Elemental Evil supplement, p. 18; also XGtE)) with cold breath (dmg 4d8 15' cone, recharge on 5-6). How would you scale down a spell which is between 3rd or 4th level in power to fit your Druid's at will power, and fit within the limits on spell casting in Wild Shape? Cold Breath is between Snilloc's Snowball Storm(2)(dmg 3d6, 5' sphere)and Cone of Cold(5)(dmg 8d8, 60' cone). As @Miniman points out in his answer, you can see how fast these at will abilities can get overpowered.
The Balance Problem
We'll look at Worg and Death Dog (Monstrosities) (Canine examples because I love my dogs. :) )
CR 1/2 Worg (1 melee attack, 2d6 +3)
- Two added languages (Goblin and Worg) which could make for some
interesting interaction and infiltration scenarios.
- Advantage for rolls based on your wisdom on smell and hearing Perception checks.
- "Knock prone" ability in combat (con save DC 12 or target falls prone).
Compare that to CR 1/2:
Ape: two fist attacks, each at 1d6 +3. Passive perception will be
overwritten by your WIS and Proficiency.
Black Bear has two attacks, bite and claw, 1d6 +2 and 2d4 +2
respectively. Advantage on smell Perception checks. (WIS).
Worg might be sellable, but it has more advantages than other CR 1/2 beasts. Does the DM require a level minimum above 4 (5? 6? 8?) to Wild Shape into one? Would he apply CR/2, and make you wait for 8th level to try that form? Another point: maybe you have to encounter and defeat a Worg, not just have seen one, to take on the shape.
Caution: the Cockatrice example returns for CR 1/2 finds a creature who inflicts on a non-saving target a Hold Person with no need to maintain concentration. That's more than a 2d level spell effect, more like 3rd or 4th. A free spell applied how many times in a given combat and not on the list of prepared spells? Big power boost. At level 4 Druid, you don't have any 3rd level spells.
CR 1 Death Dog:
- Two bite attacks per round (1d6 +2 each).
- Advantage on all Perception checks.
- Advantage on saves versus: charm, blinded, deafened, stunned, frightened, knocked unconscious.
The bite contains the balance breaker. If not saved against (CON DC 12) it creates the poisoned condition (Basic rules, p. 172) and causes a disease.
If the target is a creature, it must succeed on a DC 12 Constitution saving throw against disease or become poisoned until the disease is cured. Every 24 hours that elapse, the creature must repeat the saving throw, reducing its hit point maximum by 5 (1d10) on a failure. This reduction lasts until the disease is cured. The creature dies if the disease reduces its hit point maximum to 0. (Basic Rules, p. 123)
• A poisoned creature has disadvantage on attack rolls and ability checks.
A successful melee hit creates a spell-like effect which (if there is no save) drains max hit points (1d10) from the target. This is twice per round, one per attack. The impact of this increases with the number of opponents. During a single encounter, the disease may not be that big of a deal, but during the battle the poisoned condition imposes a significant problem: disadvantage.
Death Dog is a tougher sell than Worg. (At Druid level 8, you have 4 cantrips, 3 3rd level spells and 2 4th level spells).
Could the overboost for Death Dog be balanced? Maybe. Maybe each time your bite triggers that save requirement, your DM has you burn a spell slot (second level?) That might get closer to the balance. (Hey, wait, I thought Druids were all about the balance! :-) ) A problem is that this extra work gets the game (and the other players) away from how spells usually work in D&D 5th edition.
Brown Bear. CR 1.
- Two attacks. 1d8 + 4 and 2d6 +4, bite and claw.
- Smell Perception check boost (Wis).
- No imposed condition, no free spell/spell effects. No long term effects.
These two don't compare as well as my cherry-picked examples at CR 1/2.
So much for a rough illustration of the balance problem. It is a chore for your DM and you. If you both want to put in the work, you may arrive at something usable. This might be a step beyond how far the DM is willing to go to homebrew.
Is this a challenge to the Druid Class trope? Yes. Were I the DM, I'd say "No" due to how I see Druids in the scheme of things, but I'm not your DM. Each DM has his/her own world, so you may have some room to brew.