Depends on edition, really
...given that 5E standardized the Weave to be ubiquitous outside of Realmspace, which would render the question moot. We want to look back on earlier editions, when the Realms were part of a joint cosmology.
What the Weave is not
The casting of a wizard from Toril, or anywhere in Realmspace, does not suffer outside of Realmspace (any more than anyone else's, at any rate) because the Weave is not the power of magic itself. The Weave is the local form of the conduit between magic and mage that allows a spell to tap the necessary power to be cast. It is organized and maintained by Mystra, the goddess of magic, who embodies this organization and flow.
How it works elsewhere
Other planes, by their very nature, have a very different relationship with magic than the Prime does - there's less need for a structured conduit on, say, the Ethereal Plane or the Astral Plane. There are (under 2E Planescape rules, at least) some adjustments to how magic functions plane-by-plane, of course, as a result of these different structures.
What about other worlds, though - other crystal spheres? Why, they have different kinds of conduit - different broadband or phone service providers, if you will, though the basics of "dial a number, make a call" remain intact. The motions to go through don't change, but the specifics around what you're doing might. For instance, in Krynnspace, mages learn to harness the power of one (or more!) of three moons representing the three deities of magic in that sphere. A visiting mage from Toril could still cast using this local conduit, but would not be familiar with how to enhance her spell by tapping the unique magical properties of a patron moon. Conversely, a Krynnish mage on Toril might be surprised to discover that such methods yield no special results in this strange other world.
Various Prime worlds have various kinds of conduit, various relationships with magic. Athas, for example, is a world where the source of magic is in question, and the struggle for casters on Athas is to establish a reliable conduit to some source of power. Eberron's dragonmarks and dragonshards are emergent magical phenomena from that world which would not be expected to manifest elsewhere, though their properties can be carried off-world. On the world of Aebrynis, magic is quite literally in the blood of some people.
So feel free to travel
...because your magic doesn't rely on you being somewhere the Weave is present (though if you're reliant on a different kind of conduit, like the Shadow Weave, you may find yourself weaker off-world). Anywhere there is magic, a mage of the Realms has the capacity to cast spells, with at most some adjustments for truly challenging environments such as Athas. Just be aware that a journey isn't a guarantee that the Lady of Mysteries won't be able to cut you off regardless - Mystra knows Boccob, the moon gods of Krynn, Odin, Hecate, Avani, Isis... she'll make a few calls to troll you if you really ruffle her feathers.