Eh, because people are going to be comparing the Bard to the Monk constantly, might as well throw up some raw numbers.
Levels 1 and 2: Technically skipping this, since there is no College of Dance at these levels. If you've got a strong desire to only fight as a Dancer, ask to start at level 3.
For the below levels, I am assuming zero equipment and just looking at them as a solo fighter. Yes, the Dance Bard has tremendous support capabilities, they are a Bard. This is mostly because of the "did they eat the Monk's lunch" question.
Level 3:
Monk Offense: They can attack for 1d6+dex as their action, then again as their bonus action, for 2d6+2dex as a baseline. They can spend one of three points to attack three times instead, getting 3d6+3dex. Points refresh on a short rest, and 1/day at initiative. Unarmed strike for grapple and shove can use dexterity.
Dance Offense: They can attack for 1d6+Dex as their action. Then if they are spending one of their DAILY uses of Bardic Inspiration, they can raise that to 2d6+2dex. This maxes them at the Monk standard, for three or four turns a day max. Their unarmed grapple and shove still rely on strength.
Monk Defense: Unarmored defense, they can disengage as a bonus action getting them out of trouble, and they can use their reaction to deflect attacks if they are attacked. For a single point they can disengage AND dodge, or they can turn their deflection into damage. Points refresh on a short rest, and 1/day at initiative during which time they also heal hp.
Dance Defense: Unarmored defense with charisma. This is it.
Monk Mobility: By this point the Monk has a base speed of at least 40 ft, they can dash as a bonus action. Additionally, they can spend a point to dash and disengage. Points refresh on a short rest, and 1/day at initiative.
Dance Mobility: No innate speed increase.
Level 5:
Monk Offense: damage die increased to 1d8, gained extra attack, gained stunning strike, maxes at 4d8+4dex. Also, has two more points to spend.
Dance Offense: Damage die increased to 1d8, Bardic dice refresh on short rest, allowing for more bonus action attacks.
Monk Defense: Healing die from Metabolism increased.
Dance Defense: No change
Monk Mobility: No change
Dance Mobility: No Change
Level 6:
Monk Offense: Can ignore resistance by changing damage to force damage. Monk has more points.
Dance Offense: No change
Monk Defense: No change
Dance Defense: You can use your reaction to spend a die, move half your speed, and this allows for an attack. Since it is your die, it isn't increasing offense, since you will have one less bonus action attack now. You also have the countercharm reaction to re-roll a failed charm save.
Monk Mobility: Increased speed
Dance Mobility: No Change
The only thing left for Dance Bard that can be counted in this, is Evasion at level 14, which monk got at level 7. Monk also gets up to 5 attacks at level 10 with a point spent, gains temp hp when they use patient defense, can carry allies with step of the wind, run up walls and across liquids at level 9, cancel their own charm, frighten or poisoned condition, gain resistance to all damage, gain all save proficiencies, re-roll failed saves, and get a bonus to hit, damage, and AC at 20th level.
Does this mean the Dance Bard is weak? Absolutely not. They are still a Bard, still a full-caster, and still an enormous support. Increasing Initiative, moving allies, healing, and spells, spells, spells. However, they don't replace a monk in offense, defense, or mobility.
