To start with my answer, I want to clarify that currently almost all firm wanted to all projects in the spectrum of category. Strategy houses wanted to do operation / implementation work through different entities (e.g. McKinsey with McK implementation, Strategy& with PwC) or no separate entitite (e.g. BCG, Bain) while the Ops / implementation focus consulting also wanted to do Strategy work that can lead to implementation (e.g. PwC with Strategy&, Accenture Strategy, Deloitte with Monitor, EY with Parthenon). So having experience in any of the spectrum should not affect your candidacy to a firm as they need all skills. However, what can make a difference is the perceived analytical powers demonstrated through your past work.
As an illustration strategy work involve a short burst of sprint to create a strategy for a large company with multiple division / business units, martkets and stakeholder. So prioritization skill is very important to identify what matters, strong analytical and endurance is required to generate analysis that matters from an ocean of data, Creativity is a needed to come up with strategic initiatives that is out of the box and not a me too strategy, strong selling skill VERY important as you need to convince your clients that your strategy actually sensible.
In operation consulting analytical is important but in my experience doing operations work, different industry will have slightly different process to cater (e.g. bank and factory) but the solution more or less can be similar (e.g. throughput, turnaround time, de-bottlenecking)
Implementation is actually a very important skill these days and not many strategy consultant can do it as they don't have the patience or interpersonal skill to be successful. But this usually involve PMO of specific / all initiatives in the strategic plan, drive people to work on it (in a nice way) on top of their day-to-day work and help them solutioning any issues encountered. The work usually less fast paced (but not less pressure) than typical strategy work.