I was thinking about this lately, comparing the two, seeing what the advantages were of each over the other. This what I noticed:
In terms of space/cost efficiency, a mining base seems more efficient because it does not require any propulsion systems. So, credit for credit, you can harvest & store more resources in a mining base than you can in a mining ship of the same size. So at first glance it seems that mining bases are better.
However, in order to actually get those resources you need a freighter to go fetch it. This means that there are really two limits to your mining capacity: That of the base and that of the freighter, and whichever is smaller is your real limit (meaning having 100,000 storage on the base doesn't do much good when your freighter only has 1,000 carrying capacity, unless you want to send 100 freighters to the base at a time). Those freighters also cost money, which rapidly raises the real upkeep cost of you mining bases. In this light maybe a mining ship would be your better choice?
Of course, you can also use some of that mining base efficiency to arm & protect them, as well as place Long Range Scanners on them to increase your situational awareness of the overall map. That scanning capacity is particularly important, because while you can certainly put LRS on ships, they only work when the host isn't moving. Meaning that ship-based LRS will rarely be in use while base mounters LRS will always be on. When it comes to intelligence gathering, stations are the better choice.
Now mining ships, on the other hand, lose a good bit of mining potential for the sake of mobility. The engines, the extra power, the fuel tanks, etc, all cut into the number of harvesters and storage space - at least compared to a mining base. However, that being said, that mobility advantage is pretty darn important. Unlike a base, a mining ship can simply flee when attacked, so outfitting them with weapons is moot. They really only need enough shields to survive long enough to get into warp, and enough sublight speed to escape from interdictors. When it comes to survival, a mining ship has much better odds... especially when huge fleets show up.
More importantly is that a mining ship is it's own freighter. Once it's full it will deliver the good entirely on it's own. This matters because, as mentioned before, freighters cut into your budget so "removing the middleman" will definitely help your profit margin considerably. Likewise, even though pirates may capture your mining ships, they can't raid them per se for loot. So the cost effectiveness of mining ships is actually, when looking at the larger picture, actually better than that of stations because they don't require freighters to complete the transaction. Again, mining ships do better at this.
Speed is another factor. A mining station with a large cargo capacity, ample docking bays and a steady flow of freighters need never stop harvesting. It's flow of goods is potentially uninterruptable. Meanwhile a mining ship must spend time flying to a target and flying back, time in which it is not actually harvesting anything. In this area a mining station is the better choice.
Intelligence also comes into play: A mining ship will (usually) go out and look for things that you actually need. The AI generally knows that if you're out of steel then it's time to go look for steel. A mining station, obviously, simply mines whatever is there whether you need it or not. It cannot simply pack up and move to another location as your needs change. So in this department the mining ship is clearly superior.
When you compare speed, mobility, AI and cost my thought is that both are useful but in very different ways. Mining ships are far more useful to small empires and pirates, as they don't give away your home position, are mobile and adaptable to your rapidly changing needs. For large empires with greater, steady needs and larger budgets to work with mining bases become more desirable - provided you can secure the area, of course.
And this is where I see the real strategic issue, and it really has nothing to do with your own harvesting at all: Territorial Control. First, if you're an empire you know that you can't build mining bases in another empire's territory. You mining ships don't have this restriction, however, and ignore borders with impunity & without incident. If you're a pirate (or at war) you can, however, and this changes everything.
The real purpose of the mining base is not only to secure a resource for yourself, but also to deny that resource to the enemy. Arming a mining base is pretty much essential, after all, to hold off pesky scouts, deter pirates and take down a few enemy ships even if it gets overrun and destroyed. Even the most modestly armed mining base can deter & destroy enemy construction ships and mobile mining ships with ease, and that's what really matters. When you place an armed mining station over a resource that is one fewer resource node that your rivals can access by any means.
Now you may look at your huge map and say "But there are dozens of planets with that resource, what difference will one make?" True, but I'm not talking about steel. I'm talking about rare resources like dilithium crystals, loros fruit & zentaiba fluid. Things that most certainly can be completely monopolized. Also, strategically, just because it exists doesn't mean it's accessible.
As any pirate player knows, one sure way to ruin an empire's day early in the prewarp days is to move your construction ship into their home system and build an armed mining base on their freebie caslon gas giant. Do that and now they can't get any fuel for any of their ships, and that was their only supply. From there you can either squeeze them until their ships are all out of fuel or extort the situation to be their sole supplier via simple smuggling missions.
Likewise, while your expansion planner may show ten different sources of aculon left in the game, how many of those are actually in fuel range of your target? Two? Your rivals have fuel & range limits too, plus enemies of their own. Even if you can't cut off their supply completely you can certainly make it frustratingly difficult for them to get. Simply put, an enemy who has to cross three sectors to get something is not going to have as much available as one who can mine it in their home system.
As the real-life "U-Boat Menace" of WW2 showed, hitting an enemy's supply lines can be far more effective than engaging them in battle head-on. Knock out the dilithium crystal supply and you'll be the only one with Merridian Shields. Silicon is another economic weak point, tending towards low yields & few planets, and it is needed in nearly everything.
To that end, there is a very logical military reason to build bases as they effectively expand your area of control just as much as colonies and space ports. That is not to say that mobile mining ships are not without their own virtues, especially to pirate players - they can keep you going even when you have no bases left at all (which can most certainly happen) and with smuggler neutrality can be made absolutely dirt cheap (why bother with shields or weapons on a ship that no one will ever attack?).
So those are just my thoughts on the issue of whether one should focus on building mines or just relying on mining ships. The both have their uses, but it's important to know what your goal is and choosing the appropriate one to go with for the job.