Wedge sum of circles and Hawaiian earring

The point is that topology the Hawaiian earring inherits from $\mathbb{R}^2$ is not the topology of the wedge sum of the circles which make it up. In particular, any open neighborhood of the origin in the Hawaiian earring completely contains all but finitely many of the circles, which is clearly not the case for an infinite bouquet of circles.

You have two very nice answers discussing the difference between the topologies on these spaces. However, I thought I'd mention one slightly higher-level difference between them. The fundamental group of the wedge of infinitely many circles is the free group on countable many generators, one for each circle. This is a rather uncomplicated countable group. The fundamental group of the Hawaiian earring, however, is truly bizarre. In fact, it is uncountable and has many rather complicated relations in it.

When I first learned about this, I was shocked that closed subsets of the plane could have uncountable fundamental groups.

A nice paper that discusses this (and contains a good bibliography of earlier work) is "The combinatorial structure of the Hawaiian earring group" by Cannon and Conner, which appeared in Topology and its Applications, Volume 106, Issue 3, 6 October 2000, Pages 225-271.

The point at which all the circles meet has different neighborhoods in each of the spaces you mention. In the case of a wedge of circles, the "wedge point" has contractible neighborhoods, while the corresponding point in the Hawaiian earring has no contractible neighborhoods.