Ok so whats the difference between active PU and active EQ and which combination is best for a hot DI recording? I don't plan to play live.
An active pickup requires a voltage source (battery) to work properly. It's an attempt to get the low noise of a humbucker, but boost the sensitivity to be like a single coil. An example is this guy:
http://stores.channeladvisor.com/amazingtone-music/items/item.aspx?itemid=1041423
You can hook this up to a standard passive volume/tone like a standard Stratocaster or a Precision bass. Active pickups are not very common on bass guitars, to my knowledge, but they're out there.
An active EQ uses a voltage source (battery) to run a preamp that can BOOST or cut the tone. They'll normally have a low/high or low/mid/high knob. You can have all kinds of different features here because there is a battery driving whatever EQ circuits you want (or more - Cort used to sell a guitar that had onboard distortion, chorus, and flanger effects).
A standard passive EQ is a low pass filter. On 10, the high frequency passes through fully. On 0, the high frequency is reduced. The shoulder of the frequency curve is determined by the RC characteristics of the filter (the resistance of the tone knob, and the size of the capacitor involved). There is NO BOOST on a passive EQ. This is what your standard Stratocaster or Precision bass has.
The advantage of active EQ is pretty obvious - you have boost/cut control over your tone right on the instrument. The reason that I personally do not like active EQ is that I have an expensive preamp on my amplifier and an expensive 3 channel preamp pedal - I don't need a third cheap tiny 9-volt preamp on the instrument. However, there are some nice aftermarket bass preamps, like these guys:
http://www.bestbassgear.com/j-retro-preamp.htm
http://www.aguilaramp.com/products_preamps.htm
But they have nothing to offer me except a dying battery while I'm trying to perform (see above - I already have all the preamp controls I need). Even if it dies while you're practicing or recording, you still have to dig up another battery to keep working.
In my opinion, the stock preamp on almost any bass isn't worth using [FENDER I'M LOOKIN AT YOU and your INSANE PRICE INCREASES]. They're designed 1) for low cost, 2) for low power (battery life), and 3) to fit in a tiny space. (I owned a Fender MIM Deluxe Jazz V with active EQ and it never, ever, under any circumstances, sounded right, in my opinion.)
If you're going to go bass -> DI -> recorder, there is an argument for active EQ on the bass. But in general, it would be better to go bass -> quality preamp -> DI -> recorder, especially compared to the stock active EQ on pretty much any bass. A real preamp can be used with any bass or guitar, and it isn't designed 2) for low power (battery life) and 3) to fit in a tiny space.
So that's the info. Go forth and buy whatever sounds best to you and fits your needs.