Well, I am only able to offer my two cents on this question, though I am by no means an expert. As I understand it, both definitions have tremendous overlap and it is more important as to how the phrase is being used. But for the sake of writing an application or understanding a law firm's position in the market, I can try to make sense of the definitions.
Sector focus refers to firms that have actively invested in its practice to servicing clients in an industry. As a law firm, this means providing tailored legal advice to those clients based off its expertise in the industry. Market-leading capability better references the practice strengths the firm has to offer and its particular USP when compared to competitors offering the same services. This can be noted through rankings, significant deals, top industry clients, specialist partners/teams and any evidence of investment made by the law firm to show a competitive edge in being the go-to for certain types of work.
How do both definitions fit the context of a law firm?
The example I can use here is a Holman Fenwick Willan, a firm that can be described as having a strong sector focus. The firm divides itself into six sector-based strata (Aerospace/Aviation, Construction, Commodities, Insurance, Shipping and Energy; Shipping its most prominent). It is considered an expert in these areas due to its historical focus on developing long-standing market leaders in these areas, understanding the key commercial challenges in these industries and providing tailor-made advice to such clients. The firm was founded in 1883, and built its reputation by acting for maritime clients following shipwrecks and collisions. This established identity allowed it to build strategic relationships with insurers as well. Nowadays, the firm boasts partners and associates alike, who have been trained or acquired specifically to show they understand the related risks in the noted industries. The firm even has its own team of Master Mariners (qualification which allows one to Captain any vessel anywhere in the world). This team provides an unparalleled wealth of knowledge on the operation of vessels and can provide complementary support to the legal teams to better inform them when acting for or against insurers and other potential claimants/defendants in shipwreck/collision matters.
So, in essence, the firm has a strong sector focus like many other firms, but also possesses market-leading capabilities in certain aspects of the sector that set itself apart. I don't know how helpful this so please let me know if something needs clarifying.
