I recently did a Biz class trip: DEN->LHR on United and LHR->LAX on British Airways. I am a GS; I took advantage of BA's recent status-match offer, so I also have BA Gold status. Here's a quick comparison:

IPhone App: United is 10X better. Frankly, it's better than any of the other airline apps I've seen. The BA app kept telling me: "Can't retrieve your reservations right now..."

Checkin: BA gold crushes United on this: the LHR checkin for Gold members has its own security (no lines) and goes directly into the First lounge (which you have access to). Even beats GS checkins.

Boarding: my BA plane had remote stand (i.e. third-world) boarding. I don't know how common this is, but it sucks. I haven't had this on a UA intl. flight for at least a few years.

Seats: BA's new Club World suites are slightly superior to the Polaris suites. They have lots more storage and good privacy with doors. In lie-flat mode, the BA seats are moderately more comfortable because they are wider. However, like the UA seats, they have a fairly large gap roughly where your lower back is. The mattress pad fails to cover this.

In-flight service: BA crushes UA here. The first and second meals were served as multi-course meals. The food and wine quality were both superior. The BA FAs were extremely solicitous AND PRESENT throughout the flight (unlike the UA ones, who tend to act resentful if you want anything between the first and last hour of the flight).

I just booked another RT on BA (JFK<->LHR) for only $2600. I don't see how UAL comes close.