Well I am mostly staying when I am on Business trip in Osaka at the Hilton, since it is our company hotel.
Well I stayed over the last years more 6 weeks in total, it is big, very comfortable and if you have a HH Honors Gold status, they will upgrade you to the Eexecutive Floor their lounge is very nice. The Westin is certainly the same quality as the Hilton, better at the WEstion are for sure their heavenly beds , which are better than Hilton.
But the Hilton certainly has the better location being opposite the Umeda station and being inside of a shopping mall, which unfortunately is very expensive, it is also opposite of Yodobashi, one of the biggest electronci shops in Japan.
The hilton is certainly more convenient if you want to go out in the evening and not taking a shuttle service ( which I heard the WEstin has) or a taxi.
So in the end it comes down to preference and maybe with what Hotel chain you have the better status.
It is in my opinion a matter of personal preference, since I honestly think that both hotels are about the same quality, but me personally I always liked Westin more than Hilton, but unfortunately I have to stay at the Hilton because of company policy,
but if you want to have a real treat, go to the Ritz Carlton, Osaka, Club floor level, very near the Hilton.... this is a different league, I can assure you, I loved it there.