There is a difference. Having studied these series, I cover three aspects (trying to be as brief as possible). I write with the 1960s versions foremost, including the one-hour TZ (S4).
The difference seems clearest looking at famous TZ episodes: TZ plots are contained, using high-concept to illustrate character. Some are entirely twist-based, such as “Time Enough at Last,” “Third from the Sun” and “The Invaders.” Some lack explanation, such as “Mirror Image” and “Shadow Play”; “It’s a Good Life” is acid satire; “The Masks” and “Kick the Can” are structured for reward and punishment. None of these could be OL, imo, without major revisions.
Ethos: Both shows were conceived and produced by intellectuals, and each has a clear voice, including morally. The answer from Jo452 is generally correct but, for me, a bit overstated. Reviewer Glenn Kenny (TV Guide, June 25 1994):
Even the creepiest T-Zone visions were tempered by Serling's
patented brand of liberal humanism, but, generally, no such relief was
found in the more paranoid climes of The Outer Limits.
OL is colder, more circumspect (the 1990s version became notorious for dark endings).
Typically, TZ warns of conformity and complacency, OL warns of hubris and conspiracy. With its hard-science focus, OL is less likely to address bigotry. "Nightmare" is a classic exception, although it seems closer to a TZ, as does the relatively sentimental “The Inheritors.”
Accessibility: Both were ambitious, in using: fantastic elements; psychology, philosophy and irony; and (sometimes) ambiguous or downbeat endings. Still, in the early 1960s, TZ was more accessible for American viewers. The shows may seem more similar now than then.
OL was on ABC, a perennial also-ran, and thus could afford to be more experimental (perhaps too much: it was cancelled after two years). TZ was on CBS, the "Tiffany network," and it was the popularizer, and generically diverse. 1990s OL producer Pen Densham (same TV Guide article):
Twilight Zone ... would occasionally go into a more fanciful or
surrealistic direction, (but OL) is a pure science-fiction show.
TZ themes were universal. In The Twilight Zone Companion (2nd ed., 1992), Marc Scott Zicree called TZ
possibly (the only) TV series to deal on a regular basis with the
theme of alienation.
ABC decreed every OL have a “bear,” meaning a monster or other shock. Per Videohound's Sci-Fi Experience (1997), OL "probably caused more nightmares than any other show."
In character terms: it's been said TZ stories have ordinary people in odd situations, OL is extraordinary people in odd situations. This is paralleled in the narration. For most viewers, the (on-screen) fantastic-anthology hosts were the Other, usually British (Alfred Hitchcock, Boris Karloff), or in Rod Serling's case, Jewish. They represented older cultures, and were guides to the unknown, including the future (e.g., the British being post-imperial). Whereas Serling elegantly introduced the unusual, OL's (unseen) Control Voice seemed to suggest the viewer was unusual.
Influence: TZ's is broader and deeper in both directions. OL stays within s.f./horror; TZ takes in B-movies, fables and parables, and the short story (e.g., "The Lottery," "The Monkey's Paw," O. Henry). TZ featured more prose adaptations (not always remembering credit). OL was more Shakespearean: its protagonists are admirable with a fatal flaw.
In the late 20th century, a disappointing movie with a fantastic premise would be dismissed as "a long Twilight Zone." TZ’s influence can be invisible because vast: it created the audience for Jacob's Ladder, Fight Club, Being John Malkovich, The Purge.
Tech noir seems more akin to OL: Alien, Blade Runner, Terminator, Hollow Man, Splice. Joseph Stefano (TV Guide Sept 2, 2000), described his scripts for original OL as
unsuspectingly devising what came to be called Gothic Science Fiction.
The sci-fi element gave us monsters, and Gothica provided noirish
tales and enigmatic characters
OL used scientific advances to examine morality, as do Contact and the Bourne movies. Alan Moore's Watchmen seemingly lifted a plot element from OL “The Architects of Fear.” James Cameron famously described The Terminator as a ripoff of OL (Harlan Ellison sued and won). TZ is more akin to M. Night Shyamalan, and existential thrillers like Flightplan.