Compare white LED radiated optical power: lumens vs milliwatts
The main problem here is that Mouser have included a rubbish specification from an unknown source  - nobody quotes mW flux for white LEDs.   mW are quoted for very short wavelength ble end of spectrum LEDs that are used for applications where their short wavelength results in super-low lumen figures but gives high energy per quanta - eg applications such as dental adhesive hardening. 

In this case the REAL specs for the Lumileds product can be found here - link is on Mouser page.

However, if you HAD to compare lumen with mW.

A good modern LED outputs maybe half it's total input energy as light and will have a high lumen/Watt rating - maybe 150-200 l/W.
So a 690 mW flux out LED may have around 1.4W in and at 150 l/w gives about 200 lumen.
That's inexact (and then some) but the 600 lumen LED has enough greater output that it is almost certain to have more flux ouput.

BUT you asked about "brighter" so you want lux = lumen/m^2 so you care about both cone angle (radiation angle at half brightness" AND wavelength due to lumen variation due to eye sensitivity response curve. AND you MAY care about CRI (colour rendering index) as while that's not directly related to brightness per se, the differences in perception with CRI will affect how people feel about brightness, or may.
And .... . 
It's complex :-).

Generally data sheets do give lumen and lux and radiation curves - so you generally do not have to get too too arcane.