What is the fan tip speed at maximum thrust?
Second question first, the higher the jet engine thrust, the more efficient it is (in terms of thrust for fuel rate, bang for buck).
(Source: Boeing)
On to the Trent 900 and PW6000, the big one and small one. The bigger fan will have slower RPM. From the type certificates the max permissible RPM's are 2818 (takeoff) and 6350 (undefined), respectively.
The fan diameters are 2.95 m and 1.435 m, which translate to 435 and 477 m/s. Speed of sound in dry air at 20°C is 343 m/s.
Here's the caveat, the fan RPM (%N1) at takeoff does depend on the ambient air pressure and temperature (among other operational considerations). But as you can see, the smaller fan can rotate faster—read: max permissible—because it has less mass and the centrifugal forces are more manageable.
And since the 1960's or so, the fan RPM is rarely a limiting factor for the jet engine operation, so the engine manufacturer will have designed for the optimum blade tip speeds.


N1 RPM limit

It might be necessary to limit the engine RPM in order to avoid overloading the fan blades in centrifugal force, and to control the fan blade tip speeds. This is purely an N1 limit and is not a function of altitude.
The N1 limit is rarely more restrictive than the pressure and temperature limits.

(Emphasis mine; Jet Transport Performance Methods)
So much so—as you can see below from a 777 manual—the hotter and higher the airport up to about 40-45°C, the faster the takeoff N1 is to compensate.
(Click to view)
That answers the efficiency part, and the blade tip comparison as requested.

Related: Why does lower atmospheric pressure produce higher EPR (thrust)?