Aboveground
As far as I know, there's no limit on how high you can build (technically there is one, but it's due to the way computers work). Also, you don't have to worry about aquifers flooding where you want to build or part of your base. However, a downside to an above ground fortress is that the bottom of the fortress is easier to destroy, causing your fortress to become nothing but dust.
There's a sky-ceiling cap inherent to the map, just like there's a bedrock lower limit (even if you don't count the SMR, magma sea and the little bits extra yet as an impedance, but only (I
think) the SMR and some of the less lucrative 'extra' is actually impassible on the way to the lower limit.
(Before current Caverns, when just underground rivers were a complicating thing, in the search for the odd hidden magma tubes and 'extras', I used to generate my maps specifically with double/triple depths just for down-space, and you can do the same for sky (maybe that's what you mean) but the limit is still there and not just the point where a double-unsigned integer overflows or something.)
Unless
you do it (or set it up to be doable, by building sections atop of supports, with gangplank access from grounded stairwells using bridges?) nothing non-player, yet, can destroy your aboveground fortress by destroying the lower levels. Unless something's changed that I should have heard of.
Advantages: needs no tools to build walls, only materials. The getting and the hauling of which are left as an exercise to the reader, but scrupulously blocking my rocks. Or 'rock' - I like to stick to either one particular rock (something ubiquitous like marble, or something extravagant like marble) or a limited number of key rocks (outer walls of precious marble, inner roads of gabbro/whatever, roads and rooves of slate - or whatever the geology allowa me), but either way that needs digging.
And no nasty surprises as creating your perfectly planned structure breaches cavern (or touches hot/wet rock) and forces a redesign/work-around (for which I use the approach to first drive shafts straight down, discovering and revealing such voids,
then fill in the safe gaps between with underground stuff), because the sky is free space.
Disadvantages:Materials. You're mostly going to have to mine. Unless you can clearcut enough forest, but. Unless you can do a little of each to power a glass/clay industry,. Unless you're going via the "everything is built in soap!" approach for which you need an industrial butchery-and-the-rest encampment set up. Unless you're doing it in ice (still needs digging, though if renewable you have to juggle your windows of opportunity between freezes and thaws). Unless you pure-cast obsidian (requires magma and water access and Engineering to an arbitrary degree). Unless you have an idea I haven't thought of.
Build order - very strict. Shortcuts and planned backtracks (see discussions about scaffolding) mean it is player-labour intensive, as much or more as it is for the in-game builders.
Vulnerability to the open air (flying enemies, climbing enemies, pesky wildlife, bad weathers, dwarf nauseas, etc) which you can somewhat mitigate only with more restrictions and temporary additions to the build-order.
Underground
An advantage to an underground fortress is that the bottom of the fortress is harder to access, so an invasion cannot destroy the fortress as easily. However, only miners can mine a hole for your fort. Another thing is that miners can get stuck in the hole, and may die if not attended to quickly.
There really isn't that much of a problem with only miners working. I start off with two (occasionally three, if I'm feeling the need for speed) and take any immigrant ones holding a pick (regardless of current skill, it gets trained up easily with on-the-job-training) and sometimes trade for a new pick and choose the new digger myself, but
rarely make them myself. I always want more digging done than I'm doing, but that scales with ability towards the general direction of personal madness, if I let it, so rather than going down that spiral I just prioritise better (with designation priorities
and miner-specific burrows, as I've described elsewhere, at least the way that works for me).
Advantages:Easy plan, semi-easy management.
Net resource-source, not resource-sink.
Security and safety implicit (if you quickly account for underground voids, wet spots, hot spots, etc), and if you're into Turtling then once you set up the appropriate barrier(s) at your surface entrance(s) then you don't have to worry too much about surface thieves, siegers, wildlife, traders, immigrants, your own hunters/fisherdwarves/etc...
Disadvantages:Still a lot of work, always a lot of waste, difficulty of forward planning, chance of a
very bad breach (balance against chance of a very bad deconstruct-related overground cave-in if you get your scaffolding/descaffolding wrong!), and basically everything else that makes being a dwarf FUN, if not actual !!FUN!!, that I don't need to mention.
For me, I dig in deep and hard (I increase the default inter-cavern rock thickness, because I love digging
so much!) and then use the bounty of rocks (as I find them, I
love a large marble layer, but I'll work with other clumps) and simultaneously dig a big ditch around the large aboveground structure(s) that configurably blocks off (and eventually covers) the entryway(s) and as much essential aboveground infrastructure as I feel I can grab. So two (or more) megaprojects: aboveground, belowground and into the ground.