I'll take off my bass player hat and throw on my FOH mixer hat for this one. This deals with live situations.
There are a quite a few different variables involved here , I'll hit the most common ones:
1.
A direct signal from the bass using a direct box- This is generally my preferred way of taking a direct signal from a bassist as it is usually the cleanest signal route to take. The sound is good ( here's where your higher end basses really shine) and can easily be tweaked to set in the mix well. You can get more midrange and highs to cut thru easier than with a mic (great for soloing). Probably the best and easiest choice without getting too complex.
2.
Direct out from the amp- This depends on whether your direct signal is pre or post eq (post means the eq knobs will change the sound you are sending to the board). I will not use a post eq send since the bass amp is really your bass monitor for your onstage sound and the eq settings used for that may not be what you need for the sound out front. Post DI sends are generally used for recording purposes. Pre eq sends are basically the same as a direct box, usually wired straight off the input jack. However in my experience, amp DI's are noisy and I much prefer a separate direct box.
3.
Using a mic-This option is good if you are really trying to recreate the sound coming out of your bass rig, especially if you are using effects.The sound with a proper mic (designed for low frequency replication) is usually fatter, warmer, and has more bottom than a direct signal, but used alone it may take a bit more work to get the presence of a direct signal.
When I am mixing concerts I always insist on both a direct box and a mic. My favorite mic being a Shure Beta 52. I get the bottom end of my mix with the kick drum and bass mic. I use the direct signal for the mid/top end of the bass sound to get it cutting thru the mix, and for bass solos to get the sound jumping out and in your face.
Sorry this is so long and probably may have made you more indecisive on your choice, but it is not a cut and dry answer. Hopefully this will give you some more insight and help you decide. Of course if you are running into a PA system, it comes down to the room, the quality of the PA system and the engineer's ears and skill level.
