They solve the problem at different points and they are not interchangeable, though they are frequently sold as though the expensive one is simply the better version.
What Each Actually Does
Interior drainage accepts that water will reach the outside of the wall and manages it once it arrives. A channel is cut around the perimeter of the slab, drainage tile is laid at the footing, and everything is routed to a sump basin and pumped out. Commonly $8,000–$15,000. More detail on interior basement waterproofing.
Exterior waterproofing stops water reaching the wall at all. The wall is excavated to the footing from outside, cleaned, sealed with membrane, new drainage tile laid, and backfilled. Commonly $15,000–$30,000 and up. More on exterior basement waterproofing.
When Exterior Is Genuinely Right
When the wall itself needs work from outside anyway — parging failed, membrane absent on a wall that needs it, structural repair required from the exterior face.
When the interior is impossible or unacceptable: a finished basement where demolition would cost more than digging, or too little headroom.
When water is entering high on the wall rather than at the floor joint, which points at the wall face rather than at the footing.
When Interior Is Right
Which is most of the time in this area. Water appearing along the cove joint where floor meets wall is water coming up around the footing and under the slab, and exterior membrane does very little about that. Interior drainage intercepts it exactly where it arrives.
It is also far less disruptive outside. No excavation means no destroyed landscaping, driveway, deck, patio or utility runs — and on a typical lot that reinstatement is a large hidden cost of the exterior job.
The Honest Summary
Exterior is the more thorough repair and it is quoted considerably more often than it is justified. If someone is proposing $25,000 of excavation for a cove-joint problem, ask them to explain what the membrane will do about water arriving from underneath the slab. It is a fair question and the answer is revealing.
Ranges for both are on the costs page.