Comparison between FMCW radar modulation techniques
Assuming your receiver is a "homodyne architecture (mix TX and RX together)" as stated by mkeith, the two methods (sawtooth vs triangular) will be identical assuming your scene is stationary during the measurement period. The low-pass filtered mixer output will then always be proportional to the range of the scatterers, regardless of a up or down sweep. With a I/Q receiver you should be able to distinguish between a negative and positive beat, but I can't see any benefit. Only generating a up-sweep (sawtooth) will most likely be easier to implement, but that depends entirely on the hardware.

For moving targets, we need to distinguish between 2 cases. 


Slow moving targets
Fast moving targets


where 'fast' and 'slow' is relative to the sweep-time. For a sufficiently slow moving target, the doppler shift will be negligible and you can approximate it as stationary. You can find the velocity of a slow moving object by comparing the data from multiple sweeps, again the triangular vs sawtooth makes no difference.

I belive the intention of the triangular waveform is that you can now solve the ambiguity caused by a fast-moving object. In a FMCW radar, a moving target may seem indistinguishable from a stationary one. One traditional then introduces the triangular waveform to solve this ambiguity, see e.g. this open access article, especially figure 1.

Note that this only works for a single moving target, when you have multiple moving targets stuff gets more complicated so thread carefully.

In summary: In choosing between the two waveforms, there is a special case with a fast moving object where the triangular waveform may aid in extracting velocity (or should I say: radial relative velocity between the radar and reflector) depending on the velocity and chirp-rate. But for all other cases, the distinction is mute.

I hope that helped, let me know if I should clarify any of the points.