The French 4th republic went through decolonization very differently thant the english government : When the english accepted pretty early on that they could not keep their Empire and managed to keep a lot of influence within the commonwealth, the french tried until the end to keep their colonies under their rule, which led to 2 decolonization wars, in Indochina and in Algeria. When it became clear that they lost both, they proposed, in a kind of desperate attempt, their own kind of commonwealth : L'Union Française but it was too late and the colonies that had fought for their independance didn't wanted any of it, so the idea was abandoned.

Now in spite of this terrible decolonization, France still possess a lot of economic and politic influence in their african ex-colonies, as you've noted, and this is largely due to the neocolonial policies (The Françafrique) of the early 5th Republic, by De Gaulle and especially Jacques Foccart.

Sources : La France de la IV° République Jean Pierre Rioud

Ok, I'll take a stab at this (though this is something I've not looked at for a while, so it probably won't be fantastic). Firstly, I'd point out that we might question the basic premise. After de-colonization, both France and Britain maintained ties with their former Empire's in various ways. France, for example, maintains a military presence in numerous places in Africa, and has been described as the 'gendarme of Africa'. Susan Rice, points out that Mali's request for international aid following the outbreak of conflict in 2012 "wasn't specific, but it basically said, 'Help, France.'" So, it's not like France doesn't have ties to its former colonies. Like Britain, France also has fairly notable economic ties with its former colonies as well.

However, I'm going to accept for now that Britain maintains a more closer (in the sense of 'more amicable') relationship with its former colonies.

As for the reasoning behind this, I would point towards structural and ideological factors. Ideologically, there were various differences between the French and British Empires. Martin Evans highlights a "universalist" ideal (which he essentially attributes to the French Revolution). Max Lejeune wrote in 1956:

"We want the men of Algeria to be more free, more fraternal and more equal, that is to say more French"

Evans suggests this ideology could "never countenance independence", as independence required a rejection of this mix of French values/identity. He further highlights a key difference in how Britain and France saw their Empire. This is often summarised by Prime Minister Mendes-France's famous statement "L'Algérie, c'est la France, et non un pays étranger" (Algeria is France, and not a foreign country). This is a rather different ideology than Britain took. The occupants of Britain's colonies could be British, but the colonies were not Britain. In a way, this meant that anti-colonial movements in French regions required a firmer rejection of France. By contrast, Frederick Cooper describes British colonial policy thusly:

What was held out before Africans was, it seemed, the possibility that in some unclear time, maybe a generation or more, the Gold Coast could become Canada. Colonial Secretaries in the 1940s even claimed that such a trajectory had long been British policy.

By 1952 Britain had accepted that the Gold Coast was internally governed and making it's way towards independence (which it achieved fairly shortly). While Britain did try to hold on to some colonies (particularly its most profitable ones) it also went along with de-colonization in a somewhat easier manner (particularly regarding the less profitable regions). This isn't purely ideological (it also has to do with the specific factors of each region, and Britain's financial woes), but there is a bit of an ideological aspect to it.

This ideological difference also underpins some of the structural differences of the Empire, and the different challenges, facing Britain and France after WW2. The French Empire attempted to move to a more inclusive form, and to make good on its universalist ideals (under pressure to do so, of course). A new constitution was brought into being; the Overseas Ministry commenting that "the legislature wanted to mark the perfect equality of all in public life". The colonies -- or now, as they were known, territories d'outre mer --recieved representatives in the Assembly who were involved in the creation of the constitution (note that they didn't recieve representation proportional to the population they represented). Cooper describes the character of the French Union as "schizophrenic". It had an integrative drive -- there was affirmative action, for example -- but those percieved as disruptive to this unity/integration could quickly find themselves on the recieving end of repressive violence (as in Algeria).

Cooper argues Britain didn't really face this issue. He suggests that:

The more decentralized colonial structure of Great Britain did not foster such a debate over equivalence of all subjects of the Queen.

This structural difference may also explain something of why Britain's former colonies maintained relations more amicably. With the British Empire being administrated in a less entrenched manner, decolonization could be a slightly less bitter experience (not to say this was the case everywhere ofc, and bear in mind this is comparative). Even in cases where Britain did intervene (such as Malaya), there conflict doesn't really seem to see the widespread and protracted bitterness comparable to Algeria (there are specific factors at play here; the 'hearts and minds' campaign by the British and the Japanese occupation of Malaya too). In short, I think that the underlying ideological/structural differences of the Empire's in the first place probably go a way to explaining the differences in their relations with their former colonies. It's also worth mentioning briefly that, in talking about the 'commonwealth' as a whole, you also include settler colonies. Britain has prominent settler colonies (Canada, Australia) who tend to be more amicable with the imperial centre for different (and probably more obvious) reasons.

Sources:

Cooper, Reconstructing Empire in British and French Africa
Evans, Guy Mollet's Third Way
White, Reconstructing Europe through Rejuvenating Empire
Probably some stuff that I learnt quite a while back and can't remember; I've tried to stick to what it's in those sources though.