I'd give you my reasons for liking each but a warning: these are SUBJECTIVE opinions, I'm not expressing facts! Also, these are not exhaustive lists of all the benefits by any means, just the benefits that stand out for me.

BA:
1. Lounges tend to be much better
2. You can use BA operated lounges any time as a Gold even if you're not travelling OW (this at least is a fact).
3. BA's web interface is much better
4. GCHs have exclusive access to the best seats on planes until 24 hrs before departure
5. You can use the arrivals lounge at LHR even if travelling in WT/ WT+
6. Better chances of an upgrade than others.

LH:
1. Redemption tends to be easier with more availability on the whole (though long haul flights may have been tightening up lately). The best thing for me is that it costs the same to redeem a ticket involving other Star carriers as it does to redeem for LH/OS/LX/LO operated flights. It's also much "cheaper" to redeem than BA.
2. Golds have exclusive access to the best seats on the plane. Also generally speaking and as far as I know those seats don't get released to the general public unless the flight is sold out (in contrast to BA's -24hr cut-off). Also, more often than not, if there are empty seats on the plane, they'll be next to Gold customers. These are unpublished benefits and you won't find anyone to officially acknowledge they exist but they are there.
3. 2 upgrade vouchers every time you renew your membership. These have been reduced - before the HON tier came along we used to get more, but 2 vouchers are still better than none.
4. Qualification is valid for 2 years.