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Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length; about two hours long, twice the length of a normal episode. They're each normally split into two (one-hour) parts for syndication, to make them fit better into a typical network schedule, hence the 178 number.

Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length episodes. As originally broadcast, and as they're present on official home media releases, they were each about two hours long; that makes them hard to syndicate (typical network scheduling in North America doesn't like shows longer than an hour), so re-runs of each are typically split into two parts.

This is also apparent from Wikipedia's list of the show's episodes; the tables number each episode, but the series premiere and series finale each get two numbers (click to embiggen):

Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length (i.e. two hours long) and are each normally syndicated in two parts (one hour each); that accounts for the difference.

Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length; about two hours long, twice the length of a normal episode. They're each normally split into two (one-hour) parts for syndication, to make them fit better into a typical network schedule, hence the 178 number.

Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length and are each normally syndicated in two parts; that accounts for the difference.

Patrick Stewart appeared in every episode; there's a discrepancy in how those two pages are counting.

Star Trek: The Next Generation aired 176 episodes. However two of those episodes, the premiere episode "Encounter at Farpoint" and the series finale "All Good Things...", were feature-length (i.e. two hours long) and are each normally syndicated in two parts (one hour each); that accounts for the difference.