There isn't much of a difference, and it's largely personal preference. I've used both at different times and I prefer DBM for three major reasons:
1) BigWigs doesn't (or didn't when I last used it, which was IIRC around Firelands) have the big on screen notifications like DBM does; it has "emphasized" messages but they are just normal warnings with a bigger font and maybe a screen flash, and not the big centered "BAD STUFF - MOVE" or "DEBUFF ON TANK (3)" things. BigWigs also doesn't/didn't have the "Run away little girl" emote for moving, or the new countdowns or "Beware!" emotes either. In other words, BW puts more onus on the player knowing the mechanics of the abilities, while DBM is more of an actual reminder/alert. That may or may not be a bad thing, but in the past I found using BW I would overlook mechanics because I was so used to DBM screaming at me, not just a "ding" with a slightly larger than normal font popping up like a raid warning.
2) DBM is more popular; the majority of guilds that I've been in exclusively use DBM so I use DBM as well. If I joined a guild using BigWigs I'd use BigWigs, etc.
3) DBM has a "VictorySound" addon that not only plays defeat/victory tunes, but can be configured to play special "boss music" during the fight itself.
I nearly forgot, one MAJOR reason I prefer DBM is because it also has 5-man timers/warnings (and BGs as well). BigWigs had the LittleWigs addon but last I checked that was abandoned around 4.1 and never touched since beyond adding the boss names from the 4.3 HoT heroics. I like having warnings/notices in 5-man content as well as raids.