I believe there are still two weekly boats between Belize and Puerto Cortes, Honduras: D Express from Placencia on Friday morning and the Nesymein Neydy out of Dangriga on Saturday morning. Express returns Monday and NN on Tuesday. Cost is US$55 each way. To confirm times, etc. for D-Express check with the BTIA office in Placencia -- email BTIA@placencia.com or call 501-202-4506. For NN try 501-522-0063 in Dangriga. You could also check with Mundo Maya Travel in the Marine Terminal in Belize City, 501-223-0457, as I think they'll make reservations and sell you a ticket.
There are other options. For example, you can take one of the three daily water taxis from Punta Gorda to Puerto Barrios (three different companies, with departures at 9:30, 2 and 4, around US$20), Guatemala, and then from there make your way by bus to San Pedro Sula. This is your cheapest option.
Another option is to fly from Savannah (across the lagoon from Placencia, with connections from other points) on Maya Island Air nonstop to San Pedro Sula. Fare is US$163 -- not sure if Maya does their cash discount on this flight or not. The web site suggests the flight is daily, but it's not -- email or call Maya for the current days they fly.
I believe, but haven't checked recently, that Atlantic Air is still flying three times a week from Belize City (BZE) to San Pedro Sula. Fare is in the US$150 range.
Another, fairly recent option is a shuttle van/bus from Belize City to San Pedro Sula. Last schedule I saw was that it left Belize City at 6 am Saturday and is supposed to get to San Pedro Sula at 6 p.m. that same day. Cost is US$60. Check with Mundo Maya Travel for tickets. Somebody told me they had taken it and the passengers were kind of a rough bunch, but I have no personal experience with it.
The other option is to take public buses from Belize's western border at Benque through Guatemala to Honduras.
Note that the above prices don't include exit taxes and fees. The exit tax by boat is only US$3.75, by land US$18.75 and by air up to almost US$40 if flying out of BZE.