I see that "连笔字" refers to "not lifting your pen from the paper while writing" (from ArchChinese) and 行书 refers to semi-cursive. Is there a distinction between them, or are they synonyms?
连笔字 is a kind of style, whereas 行书 is a font family.
some font families (行书、草书 etc.) all have the style of "连笔". If people's handwriting is just scribbled and does not conform to any font style, many strokes are connected, which can also be called 连笔字.
行书 is a font family. It has its own standards, not only “连笔”。
連筆, accurately called continuous lines/strokes in english, is the act of writing without lifting the pen. You could still be writing every single stroke, or writing a shortened version with less strokes like those commonly seen in cursive chinese. While you can see it with calligraphy brush writing, you could also see it in someone's daily handwriting with a pencil etc. It doesn't have to be beautiful or artistic characters, as long as the strokes you do write are connected together.
行書, accurately called walking/strolling script in English, is a type of chinese calligraphy cursive. It is almost exclusively used with brush calligraphy, and most people find it hard to read due to the severe shorthand of stroke counts many characters undergo. If you have not specially studied it to learn what each character should become you will have troubles with writing or reading it. There may be a few accepted versions of how to shorten a specific character, but they are all preestablished and very specific, changing the curve of a line or connecting two strokes that aren't usually could potentially make it a different cursive character.
A final comparison: continuous strokes is the literal act of your strokes connecting to each other, regardless of formal or ugly handwriting, writing medium, writing out every character in regular script or cursive... etc
walking script is a type of cursive that frequently combines what would be multiple strokes into single strokes, sometimes by leaving pen on the paper to connect them, but more frequently by creating a new single stroke to represent the multiple strokes. A 12 stroke character becoming a 3 stroke character as an example, you would still lift your pen fully between each of the three new strokes in a standard process.
It is possible to encounter both in the same writing, but definitely not guaranteed. While 行書 with 連筆 does exist, I think you are more likely to encounter 連筆 in semicursive or regular script chinese overall: the larger amount of strokes gives a better reason to connect lines for speed, and the overall legibility is far less hindered by the stylistic choice. Compare full cursive with continuous strokes; the legibility for regular people gets very low, except maybe on characters written very large.