Noise control in op-amp circuit design by using low spec op amps
The question is "is the extra noise you saw relevant to what you are trying to achieve"? In other words is the noise in the bandwidth of interest of your signal? If it isn't, can it still cause you problems? Also, are you sacrificing anything by using the TL051 - if not then use it but there are other considerations.

Op-amps may specify so many nano volts per root hertz but, there is also the L.F. noise (usually specified as so many micro-volts peak-to-peak between 0.1Hz and 10Hz). Was the "so-called" good device actually worse than the TL051 in this area. BTW I haven't looked so I don't know but you shouldn't make assumptions.

The other equally important noise source is due to input bias currents and these are usually in the pico-amps per root hertz range. If you are using large input/feedback resistors this noise can dominate and, you might find that the TL051 is significantly better than the AD829 in this area. Again, I haven't looked but it's always best not to make assumptions.

EDIT - the TL051 is 150x better than the AD829 for input current noise. TL051 is 0.01pA per sqrt(Hz) whilst the AD829 is 1.5pA per sqrt(Hz). This will make a significant difference if the size of resistors on inputs and feedback are 1kohm or above. At 1kohm, the AD829's input current noise becomes about the same level as its input voltage noise and at 10kohm it becomes comparable with the TL051's voltage noise.

Power supply rejection ratio is another thing to consider - if there is noise on the power supply (as there inevitably is), which device is better at rejecting it?

Common mode rejection - if the amp configuration is differential, you would naturally assume common-mode noise would be always near zero - not so, check the data sheets and you will see that C.M.R. ratio gets far worse at higher frequencies.

I'm just trying to say that it's always a more complex picture!!!