Let's see if we can find numbers to support this claim: 

Tu-144D:  
Ogival delta wings without conical camber, cranked delta. [A distinctly low set & a much simpler wing][1], a poor performer at low speed, hence the later addition of canards, with no doubt a big weight penalty. 

[Concorde production version][2]:  
Double delta (ogee/Ogival) shaped wings with conical camber, more curved. Advanced, highly refined wing design. 

Comparing the characteristics:
\begin{array}{|l|r|r|c|}
\hline
  \text{} & \text{Tu-144D} & \text{Concorde} & \text{Comments} \\ 
\hline
   \text{Passenger capacity} & 155 & 120^{[3]} \\
\hline
   \text{Fuel load [kg]} & \text{95,000$^{[1]}$} & \text{95,680}^{[3]} \\
\hline
   \text{Range [km]} & 5330^{[2]} & 7222.8^{[3]} \\ 
\hline
   \text{Cruise Mach no [-]} & 2.15^{[5]} & 2.04^{[3]} \\ 
\hline
   \text{Fuel/km [kg/km]} & 17.8 & 13.2 \\
\hline
   \text{Fuel/pass-km [kg/km]} & 0.118 & 0.110 \\
\hline
   \text{L/D (Low speed)} & ? & 3.94^{[3]} \\
\hline
   \text{L/D (Approach)} & ? & 4.35^{[3]} & \text{250 knots, 10,000 ft} \\
\hline
   \text{L/D (Mach 0.94)} & ? & 11.47^{[3]} \\
\hline
   \text{L/D cruise} & 7.1 _{M=2.15}~^{[6]} & 7.14 _{M=2.04}~^{[3]} \\
\hline
   \text{T/W} & 0.44^{[4]} & 0.373^{[3]} \\ 
\hline
\end{array}

$^{[1]}$ Plane and Pilot - TUPOLEV TU-144 - [Source][3]

$^{[2]}$ Tu-144 SST Technical specs - [Source][4]

$^{[3]}$ Wikipedia: Concorde - [Source][5]

$^{[4]}$ S. Maharajan: Design of Supersonic Transport - [Source][6]

$^{[5]}$ NASA Facts: The Tu-144LL: A Supersonic Flying Laboratory[Source][7]

$^{[6]}$ TUPOLEV TU-144 - ALL YOU NEED TO KNOW[Source][7][Source][8]

So the specific fuel consumption is very close, with the Tu-144 at 7% higher. And I've used figures for the Tu-144D version which was a big improvement over earlier versions: those could not supercruise at Mach 2, unlike Concorde, which drastically reduced their range.

From [Wikipedia][9]: 
>  It is unclear why Tu-144D's maximum achievable altitude would be lower than Concorde's even regular flight altitude, given that Tupolev's data claim better lift-to-drag ratio for the Tu-144 (over 8.0 for Tu-144D vs Concorde's 7.3–7.7 at Mach 2.x) and the thrust of the Tu-144D's RD-36-51 engines is higher than Concorde's Olympus 593 engine.
>
> The lower landing speed compared to Tu-144 is due to Concorde's more refined design of the wing profile that provides higher lift at low speeds without degrading supersonic cruise performance – a feature often mentioned in Western publications on Concorde and acknowledged by Tupolev designers as well.

summary from Tony Buttler, 'Building Concorde': 

Early on in the Concorde project, wind tunnel tests found that a wing of the right shape would generate overwing vortices at low speeds and high angles of attack. This meant a big increase in lift, which lowered takeoff and touchdown speeds. 
Small strakes on the front fuselage were found to give superior results to canards (the canards interacted with the fin in undesirable ways). A fuel transfer system was used to trim the aircraft instead of relying on aerodynamic trim controls.  

The Tu-144 relied on retractable canards for low speeds, while BAC had found a superior solution. 

**Conclusion**: 
My impression is that the basic aerodynamics of the Concorde were better. Tupolev tried to compensate by optimizing the surface smoothness at the cost of maintainability (to borrow from the other answer, "high aerodynamic cleanness of TU-144 which was achieved by making the design more complicated and by lowering maintainability of the aircraft"). This meant closing off access hatches, using large complicated skin panels to reduce the number of seams, etc. 


  [1]: https://www.linkedin.com/pulse/concorde-vs-tu-144-aeronautical-engineers-design-awadh-kapoor-
  [2]: https://en.wikipedia.org/wiki/Concorde#Specifications
  [3]: https://www.planeandpilotmag.com/article/tupolev-tu-144/
  [4]: http://www.tu144sst.com/techspecs/performance.html
  [5]: https://en.wikipedia.org/wiki/Concorde
  [6]: https://www.sjsu.edu/ae/docs/project-thesis/Seruvizhi.Maharajan.S14.pdf
  [7]: https://www.nasa.gov/wp-content/uploads/2021/09/120315main_fs-062-dfrc.pdf
  [8]: https://www.key.aero/article/tupolev-tu-144-all-you-need-know
  [9]: https://en.wikipedia.org/wiki/Tupolev_Tu-144