As somebody that worked for a competitor and knows a lot about both of these companies, I can't say for certain what company is going to "win" local services but I can promise you both of these companies will lose (or I guess you could say Angie's List has now already lost).
Angie's List went from a membership based review service and tried to unsuccessfully turn it into a marketplace which never worked. In the process they didn't invest in their most valuable asset which was high quality reviews of service pros and a lot of consumer trust. Maybe they are still the best right now for reviews, but that is going to change very soon and this acquisition will just accelerate that.
HomeAdvisor is just a sales and marketing company (which has been around since 1999 as its old name, ServiceMagic). They aren't a technology company. They have huge teams of sales people calling pros and selling them on buying consumer leads. Consumers fill in their information for a job and their system just matches them with the pre-sold leads from their sales team. The experience is poor from the consumer side (you often get matched with terrible pros who aren't interested in your job) and is even worse from the pro side (they are paying to be matched for a job they don't want). Pros hate it, they churn at huge rates, which is why HA needs a big sales team. It's a business and they make money but it isn't the type of business that can grow very big.
I'm not even sure why this transaction is interesting. Angie's List has been trying to sell themselves for a long time and has gotten to the point where their stock dropped enough that this deal today is "good" that was bad a year ago. HomeAdvisor has been around for 18 years and will go the way of the Yellow Pages in 3-5 years. Just rearranging the deck chairs on the Titanic.