The Pros
It's convenient. It's not always possible or desirable to access records in person. In the UK, most records such as birth/marriage/death certificates, ships passenger lists and census returns are housed in Kew (London) or at the General Register Office (Southport). As I don't live near either, it would take me at least a full day to find just one or two records. The creation of paid genealogy sites means that I can use Findmypast to search for individuals leaving the UK by ship between 1890 and 1960 (and search the UK census returns); use 1911census to see where my ancestors were living at the time of the 1911 census, and sign up for GenesReunited to search the 1841-1901 census returns (excluding 1881, which is available for free here) and connect with fellow genealogists.
The Cons
It's not cheap. A six month subscription to findmypast costs £54.95, or £9.16 per month. It's actually more cost effective to take out a year's subscription, which is £89.95, or £7.50 per month. If you'd rather pay as you go, it's £6.95 for 60 credits. This might sound reasonable, but you'd be surprised how quickly your credits can be eaten up. GenesReunited will set you back around £35 for a six month's Gold subscription (which gives you unlimited searching of the census records), so around £70 per year.
Not everything is available. For example, while I can search for passengers leaving the UK on findmypast, the records for incoming passengers aren't available and would have to be looked at in Kew.
The Alternatives
FreeBMD. Searching for births/marriages/deaths here is free, but not all records have yet been transcribed - especially after around 1915, when there are quite big gaps in the available records. Eventually, this volunteer effort aims to have transcribed every single birth, marriage and death up until a set point.
Parish records. As I mentioned in this post, searching the parish registers for births, marriages and deaths will often be beneficial as long as you have a good idea of the areas that the events would have taken place. These can often be accessed via microfiche at local register offices for free or for a nominal fee.
Local family history centres. Here, you can find birth/marriage/death records, census returns for specific counties. There will generally be volunteer genealogists on hand to help with this. Most of my genealogy is now done at our local family history centre and local register office for offline sources, and FreeBMD to pinpoint births/marriages/deaths to make these searches easier.