[susan53]

No real difference in meaning, but possibly a difference in use. The Cobuild Dictionary sees it as slightly more formal. I'm not sure I agree. For me it has the added meaning of "slowly" - either because there was no hurry or because it was difficult. Here are some authentic examples from the Brown corpus (US English - but in this case I don't think variety makes any difference ) :

1. they ... made their way across the stormy Atlantic.
2. He turned and fled into the house and made his way up the marble stairs without once looking back.
3. Some chaps that know an area well can make their way by landmarks ... a tree here, a wash here, a ...
4. I suddenly saw just over their shoulders Jessica gracefully making her way through the crowd.
5. against a wrought-iron railing. His rage had built up as he made his way here from the second floor, ...
6. ... the end of the Second World War. With great difficulty he made his way from his native Hungary to Geneva to renew his
7. It was Mrs. Kirby, making her toilsome way along the veranda, laden with a ...
8. all day long the Caravan slowly made her way through the difficult passages.
9. ... Puerto Ricans, and rural newcomers are slowly making their way into the cities.
10. ... an emergency. However, the East Providence Rescue Squad made its way through to my home in time of desperation.
11. The purchase was effected and they made their way towards the hotel again, ...

Only 7 and 11 seem to support the Cobuild idea of formality - but they are counterbalanced by 3 which, with the use of "some chaps" is clearly informal. So I'd rule that explanation out.

Most of them seem to back up my idea of "slowly" except 2, (with 5 and 11 giving no indication of "speed") and your example. I know where you saw it on the web but wonder if it's actually authentic -??? It's certainly not in any of the best known corpora. If it is authentic there might be a context suggesting that he was reluctant to go to the police station, but I suspect it's just been invented and isn't a particularly good example.